This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
0, then the NTSM manifold (3.11) will be reached in finite time. Furthermore, the states x1 and x2 will converge to zero in finite time. In this section, the sliding mode control has been introduced, which provides the basis of chaos control and chaos synchronization to be described in the following sections. 3.3. Chaos Control In order to demonstrate the principle of the sliding mode based chaos control, a simple example is given first. Consider the Lorenz system described by the following equation:
x1 = σ ( x2 − x1 ) x2 = rx1 − x2 − x1 x3 . x = −bx + x x 3 1 2 3
(3.13)
Assume that the desired trajectories of the chaotic system (3.13) are x1ref, x2ref, x3ref, which are all smooth. The control signal u is applied in the third equation in (3.13), that is, equation (3.13) is rewritten as follows:
x1 = σ ( x2 − x1 ) x2 = rx1 − x2 − x1 x3 . x3 = −bx3 + x1 x2 + u
(3.14)
60
Y. Feng and X. Yu
The objective of chaos control is to force the variables of the chaotic system (3.14) to track the desired trajectories. A sliding mode manifold can be designed as follows:
s = a1 ( x1 − x1ref ) + a2 ( x2 − x2 ref ) + ( x3 − x3ref ) ,
(3.15)
where a1 and a2 are design parameters, which guarantee that the system on the sliding mode is stable. A sliding mode control law which makes the system (3.14) satisfy the η–reachability condition (3.3) can be designed as follows: u = a1σ ( x2 − x1 ) − a2 ( rx1 − x2 − x1 x3 ) − bx3 − x1 x2 + a1 x1ref + a2 x2 ref + x3 ref − η sgn( s ) .
(3.16) Therefore, the sliding mode control law (3.16) can realize the control of the chaotic system (3.14). The sliding mode control in the above example is the conventional method which characteristic is that the convergence of the system states to the equilibrium points is usually asymptotical due to the asymptotical convergence of the linear switching manifolds that are commonly chosen. We now design a NTSM controller, which is particularly useful for high precision control as it speeds up the rate of convergence near the equilibrium point. Theorem 1 can be used for chaos control. Consider a hysteresis-based PWL system constructed using a second-order linear system with a feedback of a hysteresis series as shown in Fig. 3.1a, which can be described as follows:
x1 = x2 , x2 = − a1 x1 − a2 x2 + a1u
(3.17)
where u is the nonlinear state feedback given by n
u = H ( x, n) = ∑ hysi ( x) ,
(3.18)
i =1
where H(x,n) is the hysteresis-series as shown in Fig. 3.1b, n is the number of hystereses. In order to guarantee the existence of the chaotic motion in system (3.17), the parameters should satisfy
a2 < 0 , 2 a2 − 4a1 < 0
(3.19)
61
Sliding Mode Control of Chaotic Systems
which gives rise to a pair of complex conjugate eigenvalues with positive real parts that result in useful oscillatory motions for generating chaos. The trajectory of system (3.17) can be considered as being governed by piecing together trajectory parts in the subspaces, centered at the equilibrium points, respectively.
(a)
(b)
Fig. 3.1. (a) Second-order linear system with a feedback of hysteresisseries. (b) A hysteresis-series (n = 4).
In order to control chaotic system (3.17), a control signal v is applied in the system, that is, equation (3.17) is rewritten as follows:
x1 = x2 . x2 = − a1 x1 − a2 x2 + a1u + v
(3.20)
If the objective of chaos control is to control the variables of a chaotic system to an equilibrium point of the system, the following theorem can be used: Theorem 3.2. For the chaotic system (3.20), if a linear sliding mode and a terminal sliding mode are chosen as equations (3.21) and (3.22), and a control law is designed as equation (3.23), the variable states of system (3.30) can converge to zeros asymptotically:
s1 (t ) = x1 + λ1 x2 ,
(3.21)
l1 (t ) = γ 1s1 p1 q1 + s1 ,
(3.22)
62
Y. Feng and X. Yu
v = v1 + v2 , v1 = ax1 − bx2 − au −
(3.23a)
1
λ1
x2 ,
(3.23b)
and
v2 = −
1
λ1
∫
t
0
q1 2 − p1 / q1 s1 + η1 sgn(l1 )]dτ , γ1 p1
(3.23c)
where λ1>0, γ 1>0, and η 1>0 are design parameters; p1 and q1 are positive odd integers, which satisfy the condition p > q . Proof. The following Lyapunov function is considered:
V1 (t ) =
1 2 l1 (t ) 2
Differentiating V1(t) with respect to time, it gets:
V 1 (t ) = l1l1
p1 p1 q1 −1 1 q1 2 − p1 / q1 s1 ( s1 + s1 ) q1 γ1 p1 p 1 q1 2 − p1 / q1 = l1γ1 1 s1 p1 q1 −1 (λ1v2 + s1 ) q1 γ1 p1 p = −η1γ1 1 s1 p1 q1 −1 l1 ≤ 0 , q1 = l1γ1
when l1 ≠ 0 , two different cases are discussed as follows: 1) s1 ≠ 0 . Since p1 and q1 are positive odd integers, satisfying
s1 p1 / q1 −1 >0, it can be obtained V1 < 0 . 2) s1 = 0 s1 ≠ 0 . According to [12] the state variables of the system will not always stay on the points ( s1 = 0 , s1 ≠ 0 ) and will continue to
，
，
cross the axis s1 = 0 in the phase plane 0- s1s1 . Therefore, l1 and s1 can converge to zero in finite time. Then, the states (x1, x2) will converge to zero asymptotically. This completes the proof.
Sliding Mode Control of Chaotic Systems
63
If the objective of chaos control is to force the variables of the chaotic system (3.20) to track the desired trajectories, for example, assuming that the desired trajectories of the chaotic system (3.20) are x1ref and x2ref, which are all smooth, the following theorem can be used: Theorem 3.3. For the chaotic system (3.20), if a linear sliding mode and a terminal sliding mode are chosen as equations (3.24) and (3.25), and a control law is designed as equation (3.26), the errors between the variable states of system (3.20) and its desired trajectories, e = [e1,e2]T=xref − x, can converge to zeros asymptotically:
s2 (t ) = λ2 e1 + e1 , l2 (t ) = γ 2 s2 p2
q2
+ s2 ,
(3.24) (3.25)
v = v1′ + v2′ ,
(3.26a)
v1′ = −λ2 ( x2 − x1ref ) + ax1 − bx2 − au + x2 ref ,
(3.26b)
and
q2 2− p2 / q2 s2 + η 2 sgn(l2 )]dτ , 0 γ p 2 2
v2′ = − ∫
t
(3.26c)
where λ2>0, γ 2>0, and η 2>0 are design parameters; p2 and q2 are positive odd integers, which satisfy the condition p2>q2. Proof. The proof of Theorem 3 is similar to that for Theorem 2. Some simulation results of chaotic system (3.20) using Theorem 2 and Theorem 3 are shown in Fig. 3.2 – Fig. 3.11. For the purpose of the simplicity, n is chosen as 1 in chaotic system (3.20). The chaotic behavior of system (3.20) can be seen in Fig. 3.2 under the case of uncontrolled. Fig. 3.3a – Fig. 3.3d are simulation results using Theorem 2. The control signal is activated after 240 seconds. It can be seen that chaotic system (3.20) converges to zeros after 240 second.. Fig. 3.4a – Fig. 3.4d are simulation results using Theorem 3. The desired trajectory is a limit cycle. The control signal is activated after 270 seconds. It can be seen that chaotic system (3.20) tracks the limit cycle.
64
Y. Feng and X. Yu
1.5
2.5 2
1
1.5
0.5
x2
x1
1 0.5
0
0
-0.5 -0.5
-1 -1 -1.5
0
50
100
150 t(s)
200
250
-1.5
300
0
50
100
(a)
150 t(s)
200
250
300
(b) 1.5
1
x2
0.5
0
-0.5
-1
-1.5 -1.5
-1
-0.5
0
0.5 x1
1
1.5
2
2.5
(c)
Fig. 3.2. Chaotic behavior in the case of uncontrolled (a) state x1, (b) state x2, (c) phase plane.
65
Sliding Mode Control of Chaotic Systems
1.5
2.5 2
1
1.5
0.5
x2
x1
1 0.5 0
0
-0.5
-0.5
-1 -1 -1.5
0
50
100
150 t(s)
200
250
-1.5
300
0
50
100
(a)
150 t(s)
200
250
300
200
250
300
(b) 0.5
1.5
0
1
-0.5 0.5
v
x2
-1 0
-1.5 -0.5
-2 -1
-1.5 -1.5
-2.5
-1
-0.5
0
0.5 x1
(c)
1
1.5
2
2.5
-3 0
50
100
150 t(s)
(d)
Fig. 3.3. Chaotic behavior with control as in Theorem 2 after 240 seconds (a) state x1, (b) state x2, (c) phase plane, (d) control signal.
66
Y. Feng and X. Yu
2.5
1.5
2 1
1.5 0.5
0.5
x2
x1
1
0
0 -0.5
-0.5 -1
-1 -1.5
0
50
100
150 t(s)
200
250
-1.5
300
0
50
100
1.5
1
2
0.5
0
0
-2
-0.5
-4
-1
-6
-1.5 -1.5
-1
-0.5
0
0.5 x1
200
250
300
(b) 4
v
x2
(a)
150 t(s)
1
1.5
(c)
2
2.5
-8
0
50
100
150 t(s)
200
250
300
(d)
Fig. 3.4. Chaotic behavior with control as in Theorem 2 after 240 seconds (a) state x1, (b) state x2, (c) phase plane, (d) control signal. 3.4. Chaos Synchronization Chaos synchronization is a special kind of chaos control. It usually appears in two chaotic systems consisting of a master system and a slave system, which are of identical structure and parameters except for different initial conditions. The chaos synchronization means that the trajectories of the slave system track that of the master system starting from arbitrary initial conditions [15, 17]. The pioneering work was done by [7]. Since then, many schemes for chaos synchronization have been proposed [5-8]. Generally speaking, two main classes of chaos synchronization schemes can be identified i.e. based on either the transverse stability or the observability. In the first case, sufficient
Sliding Mode Control of Chaotic Systems
67
conditions for the negativity of all transverse or conditional Lyapunov exponents are considered, ensuring synchronization of identical chaotic systems with unidirectional coupling. Considering the latter approach, the problem is formulated as an observation problem, i.e. the reconstruction of the system state from measurements of an output variable. Although there are many methods of chaos synchronization, sliding mode based method is an important method of chaos synchronization because of its robustness and simplicity. In this section, we will use the sliding mode control concept to realize chaos synchronization.
3.4.1. Synchronization of Chaotic System Using Observer Many chaotic systems, such as Chua′s circuit, Rössler system, Lorenz system, and so on, can be expressed one of the following form consisting of linearities and nonlinearities in [10]: x (t ) = Ax (t ) + bf ( x ) + D , (3.27a) or x (t ) = Ax (t ) + Bf ( x ) + D ,
(3.27b)
where x∈Rn is the state vector; A∈Rn×n, b∈Rn×1,B∈Rn×m, D∈Rn×1 are constant real matrices, n≥m; f(x): Rn→R is a nonlinear scale function; f(x): Rn→Rm is a nonlinear vector function. Another chaotic system reconstructed is similar to the master chaotic system, named slave chaotic system: x (t ) = Ax (t ) + bf ( x ) + D + u
,
(3.28a)
x (t ) = Ax (t ) + Bf ( x ) + D + u
,
(3.28b)
or
where x~ ∈ R n , A, b, B, and D are the same as (3.27); control u is used to realize the synchronization of the system (3.27) and the system (3.28). u x , y ) , where y∈Rp and can be adopted two cases: u = u( x~, x ) or u = u( ~ p≥m, is the output of system (3.27). In the former, the chaos synchronization is merely a control of the nonlinear system. For the later case, the chaos synchronization is a more challenging task.
68
Y. Feng and X. Yu
In order to realize chaos synchronization only using the output signal y(t), one important method is to regard the slave system as an observer of the master system [10]. But the existing methods have some disadvantages. The first is that the chaotic systems used for synchronization are ideal. The second is that synchronization of chaotic systems with multi-nonlinearities was not well done. This section addresses these problems. A robust observer is proposed in [14] and used to realize the chaos synchronization. Since the method of the chaos synchronization makes the systems robust, it has significant meaning for practical applications. 3.4.2. Chaos Synchronization Using a Robust Sliding Mode Observer Consider a class of uncertain chaotic systems: x (t ) = Ax (t ) + Bf ( x ) + D + ξ (t , x )
, (3.29) where A, B, D and f(x) are the same as that in (3.27); B is of full rank matrix; ξ(t, x) = Bδ(t, x) denotes the system uncertainties, δ(t , x ) ≤ r2 + α (t , y ) , where r2 is a known positive constant and α is a known function. Because the regions of the chaotic attractors are kept in a finite space, the states of the chaotic systems are bounded, that is x ≤ m , where m is a positive real constant. For the nonlinear function f(x), which consists of the states of the chaotic system, therefore, it holds f ( x ) ≤ r1 , where r1 is a known positive constant. 3.4.3. Implementation of Chaos Synchronization Based on the Walcott-Zak observer, a new robust sliding mode observer is designed, which avoids the strict conditions, so the design can be simplified. The synchronization signal is chosen as: y (t ) = Cx (t ) , (3.30) where y(t)∈Rp is the output of the chaotic system (3.29) and p≥m; C∈Rn to be determined later. For the master chaotic system (3.29), a robust sliding mode observer is proposed as follows:
69
Sliding Mode Control of Chaotic Systems
xˆ = Axˆ (t ) + Bf ( xˆ ) + D − G (Cxˆ − y ) + Bv ( xˆ , y )
,
(3.31)
where xˆ (t ) ∈Rn represents the system states; A, B and D are the same as that of (3.27); G∈Rn×1 is a constant design matrix, and v ( xˆ , y ) ∈Rm is the control input. The observer (3.31) is adopted as the slave system for chaos synchronization. Define the error variable e(t) as: e (t ) = xˆ (t ) − x(t ) . The error system is obtained from (3.31) and (3.29) as follows: e(t ) = A0 e (t ) + B( f ( xˆ ) − f ( x ) − δ(t , x )) + Bv
, (3.32) where A0=A − GC. For the sliding mode observer (3.31), a sliding mode manifold, which is the function of the output y(t) of the master chaotic system (3.29) and the variable xˆ (t ) of the slave system (3.31), is designed as the following form: s = Me = FCe = Fe y = F (Cxˆ − y )
, (3.33) where M∈R , F∈R , M=FC, the vector C is the output matrix of system (3.29). So the design of the sliding mode manifold (3.33) depends on the design of parameter matrix M. Define e = [e1T e2T ] , e1∈Rm, e2∈Rn-m, and rewrite the error system (3.32) as the following decomposed form: m×n
m×1
e1 (t ) = A011e1 (t ) + A012 e2 (t ) + B1 ( f ( xˆ ) − f ( x ) − δ(t , x )) + B1v
, (3.34a)
and e2 (t ) = A021e1 (t ) + A022 e2 (t )
. (3.34b) And rewrite s(t) (3.32) as the decomposed form: s = M1e1 + M 2 e2 , where M=[M1 M2], M 1∈Rm, M2∈Rm×(n-m). We have the following result [16, 18].
Theorem 3.4. For the observer (3.31) of system (3.29), the following two conditions are satisfied, the chaos synchronization between systems (3.31) and (3.29) is guaranteed: 1) if the sliding mode manifold is chosen as (3.33) and the control strategy v is designed as follows: v = vl + v n
,
(3.35a)
vl = − f ( xˆ ) ,
(3.35b)
70
Y. Feng and X. Yu
and vn
(s =−
T
T
MB )
sT MB
( r1 + r2 + α (t, y) + β ) ,
(3.35c)
where r1 and r2 are known positive constants. β and η are positive scalars, β>0, 0<η <1. 2) The matrix G makes A0 in (3.32) a Hurwitz matrix, the matrix M makes A022 − A021 M1−1 M 2 a Hurwitz matrix, while the following condition is satisfied: λmax ( M T MA0 + A0T M T M ) ≤ 0 , (3.36) where λmax(As) presents the maximum eigenvalue of As. Proof. Consider the following Lyapunov function: V (t ) =
1 T 1 s s= s 2 2
2
=
1 1 ( Me )T Me = eT M T Me . 2 2
The derivative of V(t) along the error system (3.32) can be obtained: V (t ) = sT s = e T M T Me = eT M T M ( A0 e + B ( f ( xˆ ) − f ( x ) − δ(t , x ) ) + Bv ) =
1 T e ( M T MA0 + A0T M T M )e + sT MBf ( xˆ ) − sT MBf ( x ) + sT MBδ(t , x ) + sT MBv 2
From (3.36), the above equation becomes: V (t ) ≤ sT MBf ( xˆ ) − sT MBf ( x ) + sT MBδ(t , x ) + sT MBv
. By substituting the sliding mode control strategy (3.35) into above equation, it can be obtained: V (t ) ≤ sT MBf ( x ) + sT MBδ(t , x ) + sT MBvn
≤ sT MB f ( x ) + sT MB δ(t , x ) + sT MB ( r1 + r2 + α (t , y ) + β ) ≤ − β sT MB ≤ − β MB s
that is V (t ) ≤ −γ s < 0 for s ≠ 0 , where γ = − β MB > 0 . Therefore, the error system (3.32) will reach sliding mode manifold surface s=0 in a finite time tr ≤ s (0) γ . After the error system (3.32) reaches the sliding mode manifold surface s=0, the dynamic behaviour of the error system (3.32) will depend on the linear sliding mode manifold (3.33), that is: e1 = − M1−1 M 2 e2 .
(3.37)
Substituting (3.37) into the error system (3.35a), the reduced-order dynamic system of error system (3.32) can be obtained:
Sliding Mode Control of Chaotic Systems
e2 (t ) = ( A022 − A021 M1−1 M 2 ) e2 (t ) = AM e2 (t ) .
71
(3.38)
Designing the sliding mode parameter matrix M to make AM a Hurwitz matrix, then the system (3.38) will be asymptotically stable. That is, after reaching the sliding mode manifold surface s=0, the error system (3.32) will converge to the equilibrium point e = 0 asymptotically. 3.4.4. Multi-dimensional Signals Transmission via One Signal Transmission Channel Chaos synchronization has numerous potential applications in mechanics, laser and chemical technologies, communications, biology and medicine, economics, ecology etc [3, 4, 9]. For practical applications such as signal, image, speech processing and secure communication, it is desirable to use a smaller number of channels of signals from the master chaotic system for chaos synchronization. The challenge is how to do it via a scalar transmission channel. One effective approach is by using observers [10, 11]. Synchronization of chaotic systems with a single nonlinearity such as Chua system and Rössler system can be well done by using an observer via a single transmission channel, whereas, for the chaotic systems with multi nonlinearities such as hyperchaotic Chua system and Lorenz system, synchronization can not be realized by using a single signal channel. In order to realize chaos synchronization via one signal transmission channel, a method to transmit multi dimensional signals plus a synchronization signal of the master chaotic system using only one single transmission channel is proposed in [14] based on the idea of time division multiplex. The functional block diagram of the transmitter is shown in Fig. 3.5. There are three tasks in the transmitter. The first is to combine multi dimensional signals into one signal to be transmitted in one single transmission channel. The second is to add the bias level series, εi, i=1,…,p, to the transmission signal to distinguish each signal from the amplitudes. The third is to add another information to the transmission signal to represent the synchronization pulse signal. The section proposes a hardware structure as shown in Fig. 3.6. The transmission signal yε(t) is inputted to the two blocks: level adjustor and level comparator. The level adjustor is used to eliminate the bias level series, εi, i=1,…, p, from the transmission signal yε(t). While the level
72
Y. Feng and X. Yu
comparator is used to extract the sampling pulse fsyn from the transmission signal yε(t). The signal fsyn is the reconstructed sampling signal of fclk. y1
k1
y2
k2
yε1 +
ε1 . . .
yp
kp
K1 f1
yε 2
K2 f2
+
yε∗ (t )
ε2 Kp
yε p fp
+
εp pluse generator
. . .
Fig. 3.5. The block diagram of the transmitter. Zero-order Hold
Level Adjustor
∗
yε (t )
. . .
y1* (t )
Gh(s)
y h1 (t)
Zero-order Hold
f h1
y 2* (t ) fh2
Gh(s)
yh2 (t)
f hp Zero-order Hold
Level Comparator
y *p (t )
Gh(s)
y hp (t)
Fig. 3.6. The hardware structure of the receiver. 3.4.5. Synchronization of Chaotic Systems with Multi-nonlinearities The principle of the synchronization of uncertain chaotic systems with multi-nonlinearties by using observers is shown in Fig. 3.7. In the transmitter terminal, y1(t), y2(t), … , yp(t) represent p output signals of the master chaotic system, which are to be transmitted. fclk is the sampling pulse, which frequency must satisfy the Shannon's sampling theorem, while in the receiver terminal, yh1, yh2, … , yhp(t) represent p reconstructed output signals of the master chaotic system. It should be
73
Sliding Mode Control of Chaotic Systems
noticed that there is only one signal transmission channel between the transmitter and the receiver. y1(t)
yh1(t)
y2(t)
Demux
. . .
yh2(t)
yε∗ Mux
Master chaotic system
Slave chaotic system
. . .
yhp(t)
yp(t)
Fig. 3.7. Synchronization of uncertain chaotic systems. The example is hyperchaotic system formed by a pair of coupled Chua′s circuits in [10]: x = Ax + Bf ( x ) + ξ (t , x ) + D = Ax + B ( f ( x ) + δ(t , x ) ) + D
where D=0, 0 −α (b + 1) 0 −α (b + 1) 0 0 A= 1 0 0 1 0 0 −α 0 0 B= 0 0 0
0 0
0
0 10
0 −β 1 −1 − m
0 m
0
0
−1
0
0
−β
0 −α 0 , 0 0 0
f (x ) f ( x) = 1 1 , f 2 ( x4 )
and
α
0 0 0 , 0 1 0
,
(3.39)
74
Y. Feng and X. Yu
0.01sin(3π t ) δ(t , x ) = , 0.01sin(4π t ) where f1 ( x1 ) = (a − b)( x1 + 1 − x1 − 1 ) / 2 ; f 2 ( x4 ) = (a − b)( x4 + 1 − x4 − 1 ) / 2 ;
α=10, β=14.87, m=1, a = −1.27, b=−0.68. The output matrix of the system (3.41) is designed as: 0 0 0 1 0 1 . C = 1 0 1 0 0 0 According to Theorem 4, A0 is a Hurwitz matrix, choose the spectrum of the matrix A0 as {−9, −8, −6, −20, −15, −7, }, the matrix G can be calculated: T
25.1798 −14.3051 30.9801 18.7690 −0.2063 52.9225 G= −27.2248 38.2535 29.3055 −7.6269 29.6791 28.7371 .
F is chosen as: F=diag{0.005, 0.005}. According to Theorem 4, the parameters are designed as: r2=0.9, r3=0.2, α(t, y)=0, β=0.2, ε1=8, ε2=0, k1=k2=1, T=1ms. The simulation results are shown in Fig. 3.8 - Fig. 3.10. Fig. 3.8 depicts the two output signals of the master chaotic system to be transmitted to the slave chaotic system, y1 and y2. Fig. 3.9 shows the one channel transmission signal yε contains two signals, y1 and y2, plus a synchronization signal. Fig. 3.10 shows phase trajectory of the hyperchaotic Chua′s circuit state variables x1, x2 and their estimates xˆ1 , xˆ 2 ; Fig. 3.11 presents phase trajectory of the hyperchaotic Chua′s circuit state variables x2, x3 and its estimates. 4 y1 y2
3 2 1 0 -1 -2 -3 -4
0
5
10
15 Time(sec)
20
25
Fig. 3.8. Two outputs y1 and y2.
30
75
Sliding Mode Control of Chaotic Systems 12 10 8 6 4 2 0 -2 -4
0
5
10
15 Time(sec)
20
25
30
Fig. 3.9. One transmitted signal. 4 master system the observer
3 2 1 0 -1 -2 -3 -4 -3
-2
-1
0
1
2
3
4
Fig. 3.10. Phase trajectory of (x1,x2) vs.( xˆ1 , xˆ 2 ). 5 master system the observer
4 3 2 1 0 -1 -2 -3 -4 -5 -4
-3
-2
-1
0
1
2
3
4
Fig. 3.11. Phase trajectory of (x2,x3) vs.( xˆ 2 , xˆ3 ).
76
Y. Feng and X. Yu
3.5. Conclusions In this chapter, the sliding mode based chaos control has been discussed. Three methods of sliding mode methods, namely, traditional linear sliding mode, terminal sliding mode and nonsingular terminal sliding mode, have been used in chaos control. The advantages of the sliding mode based chaos control can be summarized as follows: it is robust to system uncertainties, suitable for realization, and simple to calculate. Furthermore, a robust sliding mode observer based chaos synchronization method has also been described. Based on the time division multiplex technique, a transmission method of the synchronization signals from the master system to the slave system via one single transmission channel has been designed. The advantages of this method are as follows: (i) multi-dimensional signals can be transmitted using the single channel so that chaos synchronization can be realised more efficiently without incurring significant costs of additional equipment/devices for transmitting more than one-dimensional signals; (ii) it is robust to a class of chaotic systems including internal parameter uncertainties and/or the external disturbances (iii) it is a systematic design method of chaos synchronization; (iv) it can be used for highly chaotic systems.
References [1] G. Chen and X. Dong, From Chaos to Order: Perspectives and Applications. (World Scientific, Singapore, 1998). [2] W. D. Ditto, S. N. Rauseo and M. L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65, 3211-3214, (1990). [3] M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization. (World, Singapore, 1996). [4] K. S. Halle, W. C. Wu, M. Itoh and L. C. Chua, Spread spectrum communication through modulation of chaos, Int. J. Bifurcation Chaos 3, 469-477, (1993). [5] M. Itoh, W. C. Wu and L. C. Chua, Communication systems via chaotic signals from a reconstruction viewpoint, Int. J. Bifurcation Chaos 7, 275-286, (1997). [6] L. J. Kocarev and K. S. Halle, Experimental demonstration of secure communication via chaotic synchronization, Int. J. Bifurcation Chaos 2, 709-713, (1992). [7] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64, 821-824, (1990). [8] T. L. Carroll and L. M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst. 38, 453-456, (1991).
Sliding Mode Control of Chaotic Systems
77
[9] L. O. Chua, M. Itoh, L. Kocarev and K. Eckert, Chaos synchronization in Chua’s circuit, J. Circuits Syst. Computers 3, 93-108, (1993). [10] D. Cafagna and G. Grassi, Synchronizing hyperchaos using a scalar signal: a unified framework for systems with one or several nonlinearities, APCCAS 1, 575580, (2002). [11] G. Grassi and S. Mascolo, Nonlinear observer design to synchronize hyperchaotic systems with a scalar signal, IEEE Trans. Circuits Syst. 44, 1011-1014, (1997). [12] Y. Feng, X. Yu, and Z. Man, Nonsingular adaptive terminal sliding mode control of rigid manipulators, Automatica 38, 2159-2167, (2002). [13] Y. Feng, X. Han, Y. Wang and X. Yu, Second-order terminal sliding mode control of uncertain multivariable systems, International Journal of Control 80(6), 856862, (2007). [14] Y. Feng, X. Yu and L. Sun, Synchronization of uncertain chaotic systems using a single transmission channel, Chaos, Solitons and Fractals 35(4), 755-762, (2008). [15] Y. Feng, L. Sun, and X. Yu, Finite time synchronization of chaotic systems with unmatched uncertainties, The 30th Annual Conference of the IEEE Industrial Electronics Society Busan, Korea, (November 2-6, 2004). [16] Y. Feng, L. Sun and X. Yu, Synchronization of a class of uncertain chaotic systems via observer, Dynamics of Continuous, Discrete and Impulsive Systems 13(c), 7787, (2006). [17] F. Han, L. Sun, Y. Feng, P. Wang and X. Yu, Synchronization of the Duffing oscillator by using terminal sliding mode control, Proceedings of The IEEE Region 10 Technical Conference on Computer, Communication, Control and Power Engineering (IEEE TENCON) 3, 1362-1365, (2002). [18] Y. Feng and L. Sun, Chaotic system synchronization using a robust sliding mode observer, International Conference on Dynamics, Instrumentation and Control (CDIC) Nanjing, China, (August 19-20, 2004). [19] Y. Wu, L. X. Yu and Z. Man, Terminal sliding mode control design for uncertain dynamic systems, Systems and Control Letters 34, 281-288, (1998). [20] X. Yu and Z. Man, Model reference adaptive control systems with terminal sliding modes, International Journal of Control 64, 1165-1176, (1996). [21] Z. Man and X. Yu, Terminal sliding model control of MIMO linear systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 44, 1065-1070, (1997).
This page intentionally left blank
Chapter 4 A NEW TWO-STAGE METHOD FOR NONPARAMETRIC REGRESSION WITH JUMP POINTS
C.Z. Wu School of Mathematics and Computer Science, Chongqing Normal University, Chongqing, 40047, China. [email protected] C.M. Liu* and K.L. Teo† Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA, 6102, Australia. * [email protected] † [email protected] Q.X. Shao CSIRO Mathematical and Information Sciences, WA, 6104, Australia. [email protected] In this chapter, a two-staged method is presented for nonparametric regression with jump points. After the rough location of all the possible jump points are identified using the existing efficient kernel method, a smoothing spline function is used to approximate each segment of the regression function. A time scaling transformation is derived so as to map the undecided jump points into fixed points. This approximation problem is formulated as an optimization problem which can be solved by many existing techniques. The method is applied to several examples. The results obtained show that the method is highly efficient.
79
80
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
4.1. Introduction Statistical modelling generally assumes smoothness and continuity of the phenomena of interest. However, some phenomena may experience sudden or sharp change. For example, groundwater table may undergo drastic changes in very short periods of time [15] due to sudden changes in environment, such as land clearing. When we study the portfolio management, the amount of stocks of a particular investor can be viewed as experiencing a jump when he/she purchases or sells his/her stocks. This sudden change is reflected as a jump in visual display. Without considering these jumps, we may make an serious error in drawing inference about the process under study. It is clearly important to estimate both the number of jumps and their locations and magnitudes. Problems related to regression with jump points have been addressed and investigated for more than two decades. [11] proposed an algorithm to estimate regression functions when discontinuities are present. [20] used one-sided moving average to find the locations of jumps, [9] used Fourier analysis for jump detection. [8] made use of pairing pattern and linear filter to develop a discontinuity detector for the purpose of detecting the jumps and their magnitudes. [13] estimated the locations and magnitudes of the jumps by the boundary kernels. [19] used the kernel method to estimate both the locations and magnitudes of the jumps. [18] used wavelets to provide a useful procedure for the detection of the jump points and their magnitudes. [14] proposed a two-stage estimation scheme for the jump locations, where the asymptotic properties are also studied. Most of the estimators mentioned above are of the kernel type. The key idea is to investigate the difference between the estimators of the left- and right-hand side limits for the unknown regression functions. On this basis, the locations and magnitudes of the jumps can be estimated by using the maximization argument of the jump points estimation. However, the overall fittings obtained using these methods are not very satisfactory at around jump points and around end points. At the same time, the spline method was also applied for detecting the jump points. [7] used linear splines to estimate discontinuities of the unknown regression functions. [12] applied the evolutionary algorithm to locate the optimal knots.
81
Nonparametric Regression with Jump Points
These methods are computationally expensive. In this chapter, a twostage method is developed for detecting the number of jump points, their locations and magnitudes as well as finding the spline representations for the approximation of the unknown regression function with jump points. The following notations are used in this chapter. Let X be the response variable with respect to variable t . For a given set of observations {xi ,ti } , i = 1, 2, , n , the regression function is written as: xi = m ( ti ) + ε i , i = 1, 2, , n (4.1) where m is an unknown function defined on
[ 0, T ]
,
{ε i }
i = 1, 2, , n , are independent and identically distributed N ( 0, σ
2
,
)
normal random variables. For each i = 1, 2, , n , {ε i } represents the variation of xi around m ( ti ) . Without loss of generality, we can let
T = 1 . Otherwise, a re-scaling of T can be used. To form a segmented regression, we write m ( t ) as: m ( t ) = ml ( t ) , for τ l −1 < t ≤ τ l , l = 1, , N ,
(4.2)
where τ 0 = 0 and τ N = T , where τ i , i = 1, , N , are jump points. This setting covers the case of the change of mean values (step changes) as a special case by choosing ml ( t ) as constants. Note that the form of the regression ml ( t ) is usually unknown, and hence are nonparametric. It is well-known that spline functions are effective for approximating nonparametric regressions if they are smooth. However, the smoothing spline does not work well directly for regressions with jump points, since the smoothing spline is rather sensitive to the location of jump points. Some difficulties in jump point estimation have been demonstrated in [6] for linear regression with only one jump point. Intuitively, the jump points should be identified before applying the smoothing spline to each of the individual segments defined by the jump points. This is the motivation behind our proposed two-stage method. The proposed method consists of two steps. First, we should locate all potential locations of jump points. Then, we use a time scaling transformation to transform the potential jump points and spline knots into pre-fixed points. By doing this, the parameter estimation using least squares becomes an optimization problem, which can be solved by
82
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
efficiently techniques, such as the sequential quadratic approximation optimization method. Finally, the modified Akaike's information criterion is used to determine which potential jump points are real jump points. We then obtain the final model. This chapter is organized as follows. In section 4.2, we employ the kernel method proposed by [19] to locate locations of all the potential possible jump points to within respective observation points. Then, the problem formulation is given in section 4.3. section 4.4 introduces a time scaling transform, which is motivated by the one proposed in [16]. This time scaling transform is then used to derive a new time scaling transform, which maps the undecided jump points and spline knots into fixed points. The obtained optimization problem is solved by the sequential quadratic approximation optimization technique in the Optimization Toolbox within the Matlab environment. In section 4.5, we use the modified Akaike's information criterion to determine the number and locations of jump points and therefore obtain the last model. In section 4.6, several numerical examples are presented and solved by using the proposed method. Some concluding remarks are given in section 4.7. 4.2. Estimation for Potential Jump Points In statistics, the kernel method is an efficient and powerful statistical tool to detect jump points. Therefore, it is used in our two-stage method to find potential jumps. We assume that the following conditions are satisfied. 1) The number of the observation points is sufficiently large to detect all the possible jump points. That is, the number of jump points is very small in comparison with the number of observation points and the number of observation points between consecutive jump points is sufficiently large for spline fitting. 2) The jump points should be located in the interval [δ ,1 − δ ] , where δ is some small positive constant. That is, there is no jump point in the neighborhoods of the boundary points on which they become undetectable.
Nonparametric Regression with Jump Points
83
3) There is at most one jump point between ti and ti +1 , i = 0,1, , n, where t0 = 0 and tn +1 = 1 . 4) The regression errors ε i , i = 0,1, , n, are independent and identically distributed (i.i.d) random variables with mean 0 and variance σ 2 < ∞ . Let K be the kernel function with the bandwidth h . There are two popular methods to construct kernel estimators, depending on the choice of weights by either direct kernel evaluation or the convolution of the kernel with a histogram obtained from the data. Each estimator has several important advantages and disadvantages. A thorough review on this subject can be found in the paper by [2]. [4] make use of the first method to construct an estimator to detect the vicinity of the jump points. In our two-stage method, the second method is used to identify rough positions of the jump points. For the kernel function K and the bandwidth h , the Gasser-Müller estimator is defined as: n
mˆ ( t ) = ∑ xi ∫ K h ( t − τ )dτ , for t ∈ ( 0,1) , i =1 −1
si
si−1
(4.3)
where K h ( • ) = h K ( • / h ) , sn = 0 , si = ( ti + ti +1 ) / 2 , i = 2, , n − 1, and sn = 1 . Let m1 ( t ) and m2 ( t ) be two Gasser-Müller estimators obtained with the kernel functions K1 and K 2 , respectively, using the same bandwidth h . Let J ( t ) = mˆ 1 ( t ) − mˆ 2 ( t ) . (4.4) To continue, we need to analyze the value of J ( t ) . If m ( t ) has no jump point, then, under the usual regularity conditions, m1 ( t ) and m2 ( t ) are uniformly strongly consistent estimators of m ( t ) . Thus, the variation of J ( t ) would not be obvious. If m ( t ) has a jump point, then the value of J ( t ) would have an obvious change around the jump point. In Fig. 4.1., we give an example where m ( t ) has a jump at t = 1/ 2 .
84
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
Fig. 4.1. The magnitudes of J ( t ) and the regression function. Write
m ( t ) = ϕ ( t ) +ψ ( t ) ,
(4.5)
where ϕ ( t ) is the continuous part of m ( t ) , while ψ ( t ) is a step function to characterize the jumps of m ( t ) . The magnitude of
J ( t ) corresponding to ϕ ( t ) is of small order. However, the magnitude corresponding to ψ ( t ) is symmetric and convex downward in the neighborhood of the jump point if the kernel functions K1 and K 2 are chosen such that K1 ( t ) = K 2 ( −t ) , (4.6) Furthermore, if K1 and K 2 have compact support in [ −1,1] , then the widths of the neighborhoods mentioned above are no more than 2h . Based on the above discussions, we can give a numerical procedure to estimate rough locations (and hence, number) of the jump points. Algorithm 4.1. (Kernel method to detect the potential jump points) Step 1: Choose h such that h = O n −1/ 3 .
(
)
Nonparametric Regression with Jump Points
85
Step 2: Choose a nonnegative function K ( t ) with a compact support in [-1,1] and is such that
∫
1
0
0
K ≠∫ K . −1
Step 3: Let K1 ( t ) = K ( t ) and K 2 ( t ) = K ( −t ) . Calculate J ( t ) by using (4.3) and (4.4), where h is given in Step 1. Step 4: Find all the points, which correspond to local maxima of
J (t ) . The points obtained by Algorithm 4.1 are considered as potential jump points due to Step 4. In the next section, we will use a time scaling transformation to find the accurate positions of these jump points. 4.3. Segmented Regression with Constraints Suppose that τ 1 ,τ 2 , ,τ N −1 , is a set of potential jump points (obtained from the kernel method crudely). Then, the interval [ 0,1] has been partitioned into N subintervals [τ l −1 ,τ l ] , l = 1, , N . For each l = 1, , N , let the observation points contained in the subinterval [τ l −1 ,τ l ] be denoted by t1,1 , , t1, N1 , and let x1,1 , x1, N1 be the corresponding observations. The regression function is denoted by ml ( t ) in the subinterval (τ l −1 ,τ l ] where l = 1, , N , τ 0 = 0, and τ N = T . Now, we use the segmented regression to fit the segment ml ( t ) , l = 1, , N . The most widely used approach to curve fitting is least squares. If we place no restrictions on the residual sum of squares, this method is in fact an intoporlant which may be caused a rapid fluctuation. To avoid this, we incorporate a smooth penalty in the cost function. In this chapter, we will introduce the cubic spline for fitting the segment ml ( t ) . A general cubic spline basis is defined as
{1, t, t , t , (t − θ ) 2
3
1 +
3
, , ( t − θ K )+
3
}.
(4.7)
where ( t − θ k ) + = max ( 0, t − θ k ) , θ k , k = 1, , K , are the knot points. Since the smoothness is penalized in the cost, we need the estimator mˆ l ( t ) of ml ( t ) is continuously differentiable at the knot θ k ,
k = 1, , K . Write
86
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao 3
il
k =1
k =1
mˆ l ( t ) = al ,1 + ∑ al ,k +1t k + ∑ al ,k + 4 ( t − θl ,k ) + , 3
(4.8)
where θl ,1 , , θl ,il , are the pre-fixed knots contained in the l-th segment,
al ,1 , , al ,l + 4 , are the coefficients. Since ( t − θ k )
3 +
is twice continuously
differentiable, there is no any need restrictions on ml ( t ) . Define the cost functional as the following: N
Nl
N
τi
J (τ, a ) = ∑ ∑ (mˆ l (tl ,i ) − xl ,i ) + λ ∑ ∫ 2
l =1 i =1
i =1
T
τ i−1
2
(mˆ i′′(t )) dt
(4.9)
T
where τ = [τ 1 , ,τ N −1 ] , a = a1,1 , , a1,i1 + 4 , , aN ,iN + 4 , λ is the smoothness parameter. Under the least squares method with the smoothness penalty, our objective is to find ( τ , a ) such that (4.9) is minimized subject to the following constraints tl , Nl ≤ τ l ≤ tl +1,1. (4.10) For this optimization problem, the estimates of the jump locations and the optimal regression coefficients are obtained simultaneously. Let this problem be referred to as Problem (P). For the penalized parameter λ , we can choose it by the generalized cross-validation method [3]. However, here we will choose it interactively for simplicity. To solve Problem (P), we need to compute the cost. We note that the cost (4.9) is composed of two parts. Since the first part is only related to the coefficient vector a , it is easily computed. The second part is the sum of some integrals with the integral limit related to the jump points. Thus, to compute it with its corresponding gradient is hard. To overcome this difficulty, we just introduce a time scaling transformation. By this transformation, the jump points and the spline knots are all mapped into some pre-fixed points.
Nonparametric Regression with Jump Points
87
4.4. A Time Scaling Transformation Method We suppose that τ 1 , ,τ N , are N variable times in the time interval
[ 0,T ] . A time scaling transformation is introduced such that the variable times τ 1 , ,τ N , are transformed to pre-fixed times ξ1 , , ξ N , in the new time scale. This transformation was known as the enhancing transform, which was introduced by [10] to overcome the numerical difficulties in the computation of some optimal control problems. It is defined by
dt = v (s), ds
(4.11)
with initial condition
t ( 0 ) = 0,
dt = v (s), ds
(4.12)
where v ( s ) satisfies the following conditions: •
v ( s ) ≥ 0 for all s ;
•
v ( s ) is a piecewise constant on the interval (ξi −1 , ξi ] ; ξi
∫ξ v ( s ) ds = τ i −1
i
− τ i −1 ,
(4.13)
where τ 0 = 0 , ξ 0 = 0 . Now we apply this transformation to our problem such that the jump points τ 1 , ,τ N −1 , are mapped into the fixed points 1, , N − 1, θl ,1 , ,θl ,il , are mapped into and the spline knots l + 1/ ( il + 1) , , l + il / ( il + 1) , l = 1, , N . To achieve it, we just choose N ik +1
v ( s ) = ∑∑ ξ k , j χ k + ( j −1) / ( i k =1 j =1
k +1
), k + j / ( ik +1 ) )
where χ I is the indicator function, i.e.,
1, if s ∈ I , 0, otherwise.
χI ( s) =
ξ k , j , k = 1, , N , j = 1, , ik , satisfy the following conditions:
88
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
ξ k , j ≥ 0; k = 1, , N , j = 1, , ik ;
(4.14)
N ik +1
∑∑ ξ
= T;
k, j
(4.15)
k =1 j =1
ξ k , j = ( ik + 1) (θ k , j − θ k , j −1 ) , k = 1, , N , j = 2, , ik ;
(4.16)
1 1 ξ k ,ik +1 + ξ k +1,1 = θ k +1,1 − θ k ,ik , k = 1, , N . ik + 1 ik +1 + 1
(4.17)
By this transformation, we have N
τi
∑ ∫τ ( m '' ( t ) ) i =1
i
i −1
2
N
dt =∑ ∫
i
i −1
i =1
2
( m '' ( t ) ) v ( s ) ds.
(4.18)
i
Let N
Nl
J ( ξ, a ) = ∑∑ mˆ l ( tl ,i ) − xl ,i l =1 i =1
(
)
2
N
+ λ∑ ∫ l =1
l
l −1
( m '' ( t ( s ) ) ) l
2
v ( s ) ds (4.19)
T
where ξ = ξ1,1 , , ξ1,i1 +1 , , ξ N ,iN +1 . Problem ( Pˆ ) is stated as the following. Problem ( Pˆ ). min J ( ξ, a ) subject to (4.14), (4.15), (4.16) and (4.17). Clearly, we have the following theorem. Theorem 4.1. Problem (P) is equivalent to Problem ( Pˆ ) in the sense that ( τ, a ) is the optimal solution of Problem (P) if and only if ( ξ, a ) is an optimal solution of Problem (P). Furthermore, they have the same optimal cost. 4.5. Model Selection Note that the kernel method in section 4.2 tries to detect all the potential jump points. The spline approximation method outlined in section 4.3 can be used in conjunction with the time transform introduced in section 4.4 to any possible combinations of potential jump points. Suppose that the number p is the candidate of jump points. Then, the number of jump points p is required to be chosen appropriately. Several methods for choosing it have been suggested in the literature. We propose to use the
Nonparametric Regression with Jump Points
89
Akaike Information Criterion ( AIC ) [1] for the purpose of model selection. Tong [17] applied AIC for model selection in a threshold autoregression. Hurvich and Tsai [5] derived a modified Akaike information criterion AICc in small samples and claimed that the AICc dramatically reduces the bias and improves model selection. The AICc penalizes the RSS by a function of the number of free parameters and is given by
RSS N+p + , (4.20) N N + p−2 where N is the number of observations, p is the number of free parameters in the model and RSS is the first part of the cost (4.9). Note that AIC tends to overestimate the number of parameters, thus, we will use BIC as a criterion instead of AIC when the number of observation AICc = ln
points is large enough, where
BIC = ln
RSS p ln N + . N N
(4.21)
In our simulation, for those examples, where the data produced from a mathematical function, BIC will be used as the criterion as the data can be collected as much as we need. For the real data, we will use AICc as the criterion since the observation points are limited. Algorithm 4.2. T
1. Set τ = [τ 1 , ,τ N −1 ] , which is a set of candidate jump points, Γ is a set to store the solution and set it empty, AICc = 1000. m
2.Choosing candidate jump points {τ i }i =1 ⊂ τ and for given spline knots θ , use the time scaling transformation (4.11) to solve ( Pˆ ) and evaluate the corresponding AICc (or BIC ), Let Γ ' denotes the current solution which is the set of the jump points and the regression coefficients. 3. If AICc' ≤ AICc , set Γ = Γ ', AICc = AICc' , and go to Step 2. If there is no possible candidate jump points, stop.
90
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
4.6. Numerical Example To assess the performance of our proposed method, some numerical examples are presented here. For the simulated study, the data were generated by the following equation: xi = m ( ti ) + ε i , (4.22) with ti = i / n and ε i sampled randomly from a normal distribution with standard deviation σ ε . In the next numerical examples, we choose
λ = 0.1 . Example 4.1. Let n = 200, σ ε = 0.2 , and the data is produced from the function:
m ( t ) = 2 − 2 t − 0.26
1/ 5
1( t ≤ 0.26 ) − 2 t − 0.26
3/ 5
1( t > 0.26 ) + 1( t ≥ 0.78) . (4.23)
First, we use Algorithm 4.1 to detect the rough locations of the jump points. We rearrange the observation points with the given candidate jump points. Let the spline knots vector be [0.1, 0.15, 0.3, 0.5, 0.6]. We use BIC as criterion and the obtained results are that there are two jump points: 0.25225, 0.78178. The obtained BIC is -4.1056. The result is depicted in Fig. 4.2. We do the same simulation for 100 times. The results are depicted in Fig. 4.3. If we choose λ=10, then the obtained result is depicted in Fig. 4.4. From Fig. 4.4, we can see that if λ is enough, then the fitting becomes linear fitting. Thus, the parameter λ controls the gradient change rate of the splines. Example 4.2. We now apply our method to a real example, which contains the water levels (depths) in boreholes monitored irregularly over time. The data are taken from the database of Agriculture of Western Australia and have been analyzed by [15]. There are 49 observations. The observation points as well as its jump estimator by Algorithm 4.1 is depicted in Fig. 4.5. We can see that the kernel method cannot present an good jump estimator since the observation points is too little from the figure. Thus, we have to consider all the potential jump points and use
Nonparametric Regression with Jump Points
91
AICc as criterion to choose the best model. In the process of fitting, we suppose that all of the jumps are positive since the level of the groundwater does not experience drop instantly. We re-scaling the time to the new interval [0,1]. The last obtained model has 3 jump points: 0.1050,0.2727,0.4554, AICc = 4.3148. The obtained results are depicted in Fig. 4.6.
Fig. 4.2. Example 4.1. with λ = 0.01 .
Fig. 4.3. Example 4.1. with 100 times simulation.
92
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
Fig. 4.4. Example 4.1. with λ = 10 .
Fig. 4.5. Observation points and jump estimator of Example 4.2.
Nonparametric Regression with Jump Points
93
Fig. 4.6. The fitting result of Example 4.2.
4.7. Conclusion In this chapter, a new two stage method has been proposed for solving spline regression problem with jump points. First, we detect the rough locations of jump points based on the kernel method. Then, we introduce a time scaling transformation reformulate our regression problem as a nonlinear optimization problem which is easily to be solved. Also, some numerical results are presented and which are shown that our proposed method is efficiency. References [1] H. Akaike, A new look at the statistical model identification, IEEE Transaction on Automatic Control 19, 716-723, (1974). [2] C. K. Chu and J. S. Marron, Choosing a kernel regression estimators, Statistical Science 6, 404-419, (1991). [3] P. Craven, and G. Wahba, Smoothing noisy data with spline functions, Numerical Mathematics 31, 377-403, (1979). [4] I. Gigbels, H. Peter and A. Kneip, On the estimation of jump points in smooth curves, Annales Institue of Statistics and Mathematics 51, 231-251, (1999). [5] M. Hurvich, and C. L. Tsai, Bias of the corrected AIC criterion for underfitted regression and time series models, Biometrika 78, 499-509, (1991).
94
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
[6] S. Julious, Inference and estimation in a change-point regression problem, Journal of Royal Statistical Society Series A 50, 51-61, (2001). [7] J. Y. Koo, Spline estimation of discontinuous regression functions, Journal of Computational and Graphical Statistics 6, 266-284, (1997). [8] D. Lee, Coping with discontinuities in computer vision: their detection, classification, and measurement, IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 321-344, (1990). [9] F. Lombard, Detecting change points by Fourier analysis, Technomerics 30, 305310, (1988). [10] H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parameterization enhancing transform technique for time optimal control problems. Dynamic Systems and Applications 6, 243-261, (1997). [11] A. McDonald and A. L. Owen, Smoothing with split linear fits, Technometrics 28, 195-208, (1986). [12] S. Miyata and X. Shen, Adaptive free-knot splines, Journal of Computational and Graphical Statistics 12, 197-213, (2003). [13] H. G. Müller, Change-points in nonparametric regression analysis, Annals Statistics 20, 737-761, (1992). [14] H. G. Müller and K. Song, Two-stage change-point estimators in smooth regression models, Statistics and Probability Letters 34, 323-335, (1997). [15] Q. Shao and N. Campbell, Modelling trends in groundwater levels by segmented regression with constraints. Australian and New Zealand Journal of Statistics 44, 129-141, (2002). [16] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parametrization enhancing transform for constrained optimal control problems, Journal of Australia Mathematical Society B 40, 314-335, (1999). [17] H. Tong, Threshold autoregression, limit cycles and cyclical data, with discussions, Journal of Royal Statistical Society B 42, 245-292, (1980). [18] Y. Wang, Jump and sharp cusp selection by wavelets, Biometrika 82, 385-397, (1995). [19] J. S. Wu and C. K. Chu, Kernel-type estimators of jump points and values of a regression function, Annals Statistics 21, 1545-1566, (1993). [20] Y. Yin, Detection of the number, locations and magnitudes of jumps, Communications in Statistics-Stochastic Models 4, 445-455, (1988).
Section B Chaos Control for Continuous-time Systems
This page intentionally left blank
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chapter 5 CHAOS CONTROL FOR CHUA’S CIRCUITS
L.A.B. Tˆ orres∗ , L.A. Aguirre† , R.M. Palhares‡ and E.M.A.M. Mendes§ Dept. of Electronic Engineering - DELT Universidade Federal de Minas Gerais ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] The practical implementation of Chua’s circuit control methods is discussed in this chapter. In order to better address this subject, an inductorless Chua’s circuit realization is first presented, followed by practical issues related to data analysis, mathematical modelling, and dynamical characterization associated to this electronic chaotic oscillator. As a consequence of the investigation of different control strategies applied to Chua’s circuit, a tradeoff among control objective, control energy, and model complexity is devised, which quite naturally leads to a principle that seems to be of general nature: the Information Transmission Via Control (ITVC) for nonlinear oscillators. The main purpose of the present chapter is to serve as an introductory guide to the universe of Chua’s circuit control, synchronization, and mathematical modelling.
5.1. Introduction The so-called chaos advent was rekindled after the computer simulation carried out by Lorenz in the early sixties [56]. During the seventies some basic research took place which aimed at establishing new concepts concerning chaotic dynamics [63]. The eighties were strongly marked by the goal of developing tools to characterize chaos [32, 37]. Subjects of modeling [5, 36] control and synchronization [30, 42] of chaotic systems were mainly developed in the nineties. Of course, there is active ongoing research in these fields. 97
chap5
October 7, 2008
98
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Particularly, the field of chaos control and synchronization of nonlinear oscillations has attracted much attention lately [13, 20, 21]. While much has been accomplished in the realm of theory, quite understandably much less has been reported in terms of practical results [13]. One of the reasons for this imbalance is the obvious fact that more often than not ideas (theory) develop before actual implementation. Another possible cause is the natural difficulty to assemble a suitable setup for testing new ideas as opposed to simulating solutions on a digital computer. From the mid-eighties considerable effort was devoted to build simple laboratory systems that could be used to produce chaotic data. An electronic circuit that has come to be known as Chua’s circuit is to be found among such endeavors [57, 58]. Such a circuit has undoubtedly become a standard benchmark in the study of nonlinear dynamics and chaos [28]. In order to be able to take full advantage of the many benefits of Chua’s circuit in the investigation of control and synchronization algorithms, it is important that the real circuit should include devices— called actuators— to implement the control action. In order to gain flexibility, it is usually desirable that the control law be implemented on a digital computer. This will require additional real-time I/O capabilities in order to drive the actuators [73]. The advantages of a setup that includes actuators goes beyond implementing control and synchronization algorithms. Indeed, (simulated) driven Chua’s circuits are useful in a number of different ways such as, for instance, in the identification of new bifurcation scenarios [15]. The paper [73] described a laboratory setup, named Chua’s Circuit Control Prototype – PCChua, with the following features: (a) composed of a Chua oscillator furnished with three actuators thus permitting mono and multi-variable control; (b) the actuators can be driven by the analog outputs of a standard I/O-board; in order to be able to actuate fast enough (c) the I/O-board is driven by a real time program written for Linux and (d) an inductorless implementation of Chua’s circuit permits to slow down the original dynamics to just a few hertz. This enables implementing sophisticated control schemes without severe time restrictions in applications such as: design and verification of chaos control algorithms, practical synchronization studies, experimental chaos based information transmission systems, to mention a few. In the present chapter, the aim is to present practical implementations of control strategies for Chua’s circuit. A comprehensive account of this issue should discuss the Chua’s circuit implementation itself (Sec. 5.2), together with data analysis and system identification techniques used to
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
99
Chaos Control for Chua’s Circuits
assess the quality of the implemented circuit and to gain insight on the peculiarities of Chua’s circuit dynamics (Sec. 5.3). In addition, a sample of control strategies applied to the problem of Chua’s circuit control and synchronization are discussed in Sec. 5.4. 5.2. Chua’s Circuit Implementation This section is primarily focused on presenting a robust and reliable implementation of Chua’s circuit. As it will be seen next, the circuit can be easily built, which is one of the reasons for its popularity among scientists all over the world. 5.2.1. An Overview of Chua’s Circuit The circuit in its standard form, built as an autonomous oscillator, is composed by a network of linear passive elements connected to a nonlinear active component called Chua’s diode, as seen in Fig. 5.1.
!
Fig. 5.1. Chua’s circuit. R is a variable resistor used to change the coupling between the active and passive parts of the circuit. As a consequence variation of R leads to bifurcation phenomena.
The set of differential equations that describes the circuit behavior is obtained from Kirchhoff circuit laws, such as: dx/dt = (y − x)/(RC1 ) − id (x)/(RC1 ); dy/dt = (x − y)/(RC2 ) + z/C2 ;
(5.1)
dz/dt = −y/L − z(rL /L);
where C1 , C2 , L and R are the values of the passive linear elements shown in Fig. 5.1, and rL , not shown in the figure, denotes the series internal
October 7, 2008
15:34
100
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
resistance of the inductor. The static nonlinearity of Chua’s diode is the piecewise linear function given by id (x) = m0 x + 0.5(m1 − m0 ) {|x + Bp | − |x − Bp |} ,
(5.2)
where m0 , m1 , Bp ∈ R are parameters. Figure 5.2 shows a measured static nonlinearity of an implemented Chua’s diode. 0.004
0.003
0.002
i (A)
0.001
0
-0.001
-0.002
-0.003
-0.004 -8
-6
-4
-2
0 v (V)
2
4
6
8
Fig. 5.2. Experimentally obtained static nonlinearity of a Chua’s diode. The dots correspond to real data, and the continuous traces are the resulting linear regression applied to the measured data in order to reveal the piecewise affine structure of Chua’s diode curve.
As a final remark on Chua’s circuit equations, by adopting the following conventions 0 C2 t p = C1 ; t = RC2 ; 2 (5.3) z˜ = Rz; q = − R LC2 ; f (x) = Rid (x); r = rL q; it is possible to rewrite Eq. (5.1) in normalized form as: x˙ = p [y − x − f (x)] , y˙ = x − y + z˜, z˜˙ = qy + r˜ z,
(5.4)
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
101
Chaos Control for Chua’s Circuits
where x˙ = dx/dt0 , y˙ = dy/dt0 and z˜˙ = d˜ z /dt0 . There are different ways to implement the nonlinear characteristic Eq. (5.2) [46]. A robust and reliable way is by using operational amplifiers [41], as shown in Fig. 5.3. Chua’s diode
R1
R4
R2
R5
R3
Fig. 5.3.
R6
Implementation of Chua’s diode using Op.Amps. (operational amplifiers).
From the basic circuit topology shown in Fig. 5.1, many variations have been proposed in the literature, which lead to generalizations that exhibit a richer set of dynamical behaviors [19, 47]. 5.2.2. An Inductorless Version of the Circuit Chua’s circuit exhibits a broadband spectrum, but the majority of the power density is concentrated roughly around the frequency determined by the resonant sub-circuit composed by L and C2 in Fig. 5.1. Very slow chaotic oscillations are desirable in many situations where computers are employed to measure data from the circuit, run sophisticated algorithms based on these data and to take some control action or to run some file or graphical output operation. In order to slow down the chaotic oscillations, it is necessary to use larger energy storage elements, i.e. greater values for C1 , C2 and L in Eq. (5.1).
October 7, 2008
102
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
However, despite the fact that off-the-shelf capacitors cover a very wide range of capacitance values, the same is not true for commercial inductors. Furthermore, the trade-off between coil size and maximum allowed inductance leads to very large and low-accuracy inductors. One possible solution to this problem is the replacement of the inductor by a circuit that simulates the behavior of an ideal grounded inductor over several decades of frequency [71]. This circuit is, indeed, a gyrator circuit built as an impedance transformer [14, 65] as seen in Fig. 5.4. Several other configurations of inductorless Chua circuits have been recently suggested [26, 35, 62].
i(t)
R7
+
−
zy R8 R9
v(t)
C3
−
+
R10
Fig. 5.4. Op-Amp realization of the inductor simulation circuit. The voltage at z y point can be readily used to determine the “current through the realized inductor”. In this type of circuit it is mandatory that one of the terminals of the realized inductor be grounded (this corresponds to grounding the terminal of R10 ).
From Fig. 5.4, one can see that the third state; variable z in Eq. (5.1); which is the current through the inductor L shown in Fig. 5.1, can be readily determined by measuring the voltage zy shown in Fig. 5.4 and using the expression z = (zy − y)/R7 ; where y is the state variable corresponding to the voltage across the capacitor C2 . Therefore the complete state vector is composed of voltages, which is in general easier to measure. The theoretical transfer function of the simulated inductance is [65]: Z(s) =
V (s) R7 R9 R10 C3 L{v(t)} = = s, L{i(t)} I(s) R8
(5.5)
where L stands for the Laplace transform; and R7 through R10 and C3 are shown in Fig. 5.4. Obviously Eq. (5.5) suggests that the inductor is ideal since it is a pure differentiator as indicated by the zero at the origin
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
103
Chaos Control for Chua’s Circuits
of the complex s-plane. In order to investigate the limits of this configuration, a realistic SPICE model was implemented and the resulting frequency response was found to be valid for almost four decades, which easily encompasses the spectral content of the measured time-series from the circuit, in chaotic and non-chaotic regimes [73]. Following the same procedure, it was also found that the assumption of nearly zero equivalent internal resistance rL , associated to the realized inductor, is also valid in the same frequency range. This is a very important detail in the Chua’s circuit implementation because high internal resistance rL prevents the appearance of chaotic dynamics [11]. The complete circuit is shown in Fig. 5.5. A set of electronic components values that can be used in this implementation are listed in Table 5.1. For these components, in Eq. (5.1) the corresponding theoretical circuit parameters are: 0Ω ≤ R ≤ 2kΩ; C1 = 23, 5µF; C2 = 235µF; L = 42.3H; rL = 0.0Ω; m0 = -0.409mS; m1 = -0.756mS; Bp = 1.86V. Chua’s diode
Simulated Inductor
R1
R4
R
R7 C1 R2 R3
R8
C2
R5
R9
C3
R6 R10
Fig. 5.5. Implemented inductorless Chua’s circuit without actuators. In this case the reference point of the realized inductor is attached to the same ground reference of the rest of the circuit (see Sec. 5.2.3). The component values are shown in Table 5.1.
The main resonant frequency corresponding to the parameters in Table 5.1 is approximately 1.60 Hz. This is important in real-time applications because it enables the user to implement rather involved algorithms without being pressed too hard by computation time. 5.2.3. PCChua – A Versatile Experimental Platform This section describes an experimental setup, called PCChua [73], designed to be a testbed in the study of control and synchronization schemes.
October 7, 2008
104
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes Table 5.1. Possible set of electronic components that can be used in inductorless Chua’s circuit implementation. Components Value R1 , R2 220Ω R4 , R5 22kΩ R3 2.2kΩ R6 3.3kΩ R7 , R8 , R10 1.2kΩ R9 1.5kΩ R 2kΩ (multi-turn) C1 23.5µF (47µF k 47µF) C2 235µF (470µF k 470µF) C3 23.5µF (47µF k 47µF) ICs TL08x (4 Op.amps.)1 1 Supply
voltage: ±15V.
A fundamental issue in any experimental platform for control or synchronization purposes is the design and construction of devices that will actually implement the decisions made by the controller(s). Such devices are usually referred to as actuators. In the case of the PCChua platform, the following describes the built-in actuators. The PCChua setup implements the following vector differential equation ~x˙ = F(~x) + ~u(t, ~x),
(5.6)
where ~x = [x y z]> ∈ R3 is the state vector, F(·) : R3 → R3 is the Chua’s circuit nonlinear vector field, and ~u(t, ~x) = [ux uy uz ]> ∈ R3 is the vector of control inputs. Equation Eq. (5.6) can be written in detail from Eq. (5.3) and Eq. (5.4) as x˙ = p [y − x − f (x)] + ux (t, ~x), y˙ = x − y + z˜ + uy (t, ~x), z˜˙ = −qy + r˜ z + uz (t, ~x),
(5.7)
where the parameters p, q and r depend on the components used, see Eq. (5.3). From a practical point of view, the addition of the control vector ~u(t, ~x) in Eq. (5.7) can be accomplished by including in the circuit controlled voltage and current sources (actuators), as indicated in Fig. 5.6. The control
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
105
Chaos Control for Chua’s Circuits
R + Id (x)
+
x
Chua's Diode
C1
rx
-
Fig. 5.6.
z
y
C2
L
ry
-
+ -
rz
Chua’s circuit with actuators indicated.
inputs and the actuators are related by C1 ux (t, ~x), RC2 1 ry (t) = uy (t, ~x), R L rz (t) = 2 uz (t, ~x). R C2
rx (t) =
(5.8)
From Fig. 5.6 it is clear that the relative position of the voltage controlled voltage source rz in the circuit implies that a floating realized inductor (gyrator circuit) is necessary. One solution to this problem is to use an independent power source to supply energy to the simulated inductor. With this modification, the realized inductor becomes a self-contained circuit with only two external terminals. However, it is important to notice that there are alternative simulated floating inductor circuit topologies that can be employed, as pointed out in [45]. An alternative is the use of a floating voltage controlled voltage source instead of a floating realized inductor. In the PCChua, the actuators were implemented as indicated in Fig. 5.7, using the electronic components listed in Table 5.2. The current and voltage sources have fixed gains: 1V/V for the controlled voltage sources and 1mA/V for the controlled current sources. In Fig. 5.7, the adjustable offset voltages, obtained by varying R14 and R32 , are necessary to minimize the effect of small nonzero voltage levels from the external reference voltages.
October 7, 2008
106
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
R12 R17 From I/O board
R16
− R11
−
+
+
rx
R18 +15V
−15V R19 R13
R14
R15
(a) R21 R26 From I/O board
R25
− +
R20
− +
ry
R27 +15V
−15V R28 R22
R23
R24
(b) R30 +15V From I/O board
Q1
− R29
+
+
− Q2
rz +15V
−15V R31
R32
−15V
R33
(c) Fig. 5.7. Actuators circuits with fixed gain and offset adjustment. (a) Voltagecontrolled current source rx (R11 to R19 ); (b) voltage-controlled current source ry (R20 to R28 ); and (c) voltage-controlled voltage source rz (R29 to R33 ).
The external voltage sources are actually provided by a multi-purpose data and analog output board [73], which is driven by dedicated real-time software written for a modified version of the Linux operating system, called RT-Linux, schematically represented in Fig. 5.8. The multi-purpose board is used to measure the three voltages x, y and zy (Fig. 5.1 and Fig. 5.4), and to issue three control signals rx , ry and rz to the actuators in the PCChua (Fig. 5.6). The control and synchronization algorithms are executed at fs = 1kHz, but the fastest rate at which data
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
107
Chaos Control for Chua’s Circuits Table 5.2. Electronic components used in the implementation of PCChua actuators. Components Value R11 , R12 , R13 , R15 4.7kΩ R16 , R17 , R18 , R19 1kΩ R20 , R21 , R22 , R24 q 4.7kΩ R25 , R26 , R27 , R28 1kΩ R29 , R30 , R31 , R33 4.7kΩ R14 , R23 , R32 5kΩ (multi-turn) Q1 BC547 Q2 BC557 ICs TL08x (6 Op.amps.)1 1 Supply
voltage: ±15V.
Data and Control signals I/O board
RT Linux
Fig. 5.8.
PCChua
PCChua experimental setup overview.
can be acquired is fa = 100Hz, due to hardware limitations. Nevertheless, it is important to note that fs 1.6Hz, which indicates that continuoustime versions of control and synchronization strategies can be implemented with small effort. All the measured voltages in the PCChua are acquired using 12 bits of resolution. Two of the DACs used to issue rx and ry have 10 bits of resolution and the third one (used for rz ) is limited to 8 bits. In Fig. 5.9, an example of the famous double-scroll attractor from Chua’s circuit, obtained by using the PCChua experimental setup, is presented.
5.3. Chua’s Circuit Data Analysis and Modeling This section will describe a few results concerning the analysis and modeling from data of the double-scroll attractor from Chua’s circuit. Further details can be found in the cited references.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
108
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
1
y (V)
0.5
0
-0.5
-1
-6
-4
-2
0
2
4
6
x (V)
(a) 6 4
x (V)
2 0 -2 -4 -6 0
10
20
30
40
50
60
t (s)
(b) Fig. 5.9. Data collected from PCChua during 60s. (a) Measured double scroll attractor from Chua’s circuit. (b) Time-series corresponding to the voltage across capacitor C 1 (x-variable) in Fig. 5.1.
5.3.1. The Issue of Observability The key concept of observability is that of deciding if all the dynamical information of the full phase space Rm of an m-dimensional system can be retrieved by measuring one variable s(t) = h(x), h : Rm 7→ R, referred to as the observable and h, as the measuring function. Briefly, a system is observable if the full state can be found based on s(t) only. Let x˙ = f (x), with f : Rm 7→ Rm , and s(t) = h(x), with h : Rm 7→ R be a nonlinear system and a scalar observable, respectively. From s
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
109
a reconstructed space can be formed, for instance by taking a sufficient number of successive derivatives X = (s(t), s(1) , ..., s(d−1) ), where s(j) is the jth time derivative of s [25]. Therefore, there is Φs that maps the original phase space to the space reconstructed from the observable s, that is, Φs : Rm (x) 7→ Rd (X). Basically, Rm (x) will be observable from s(t) if it is possible to go back from Rd (X) to Rm (x) on a one-to-one basis, that is, if Φs defines a diffeomorphism. Observability of Rm (x) from s is therefore directly related to the existence of singularities in Φs . In this respect it would be interesting to answer the following question: which variable is best to observe the dynamics of Chua’s circuit? The analysis of the observability of the double scroll attractor is typically very challenging. One of the difficulties with this system is that the great difference in the velocities of convergence and divergence along the stable and unstable manifold. As a consequence, some definitions of observability indices, here denoted by the letter δ, are very small (≈ 10−15 ). Another potential problem is that the application of some techniques to discontinuous systems is questionable. Such difficulties seem to have been overcome in [4] where normalized observability indices, indicated by δ ∗ , were defined. The following results are obtained: δx∗ = δy∗ = δz∗ = 1, which is consistent with the fact that locally the system is linear almost everywhere, and we would expect to have “global” diffeomorphisms almost everywhere too. As the tolerance (a parameter used in the new definition) is increased δy∗ quickly diminishes, whereas δx∗ = δz∗ = 1 remain. Increasing the tolerance even further, δz∗ drops, thus giving the observability order x . z . y (see Table 5.3). Table 5.3. Normalized observability indices δs∗ for the double scroll attractor for increasing values of a tolerance. These results give the observability order x . z . y. p δx∗ δy∗ δz∗ 0.01 1 1 1 0.02 1 0.68 1 0.10 1 0 1 0.25 0.68 0 0
The results shown in Table 5.3 clearly point out that the case of the double scroll attractor is numerically badly conditioned and suggest that the best observable is x, followed by z and, finally by y. This order was confirmed by another observability test, based on data. This result, and in
October 7, 2008
110
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
particular the poorer observability properties of the observable y has been felt in the context of data sampling [2] and modeling [9] which is probably due to the fact that observing the system from y all symmetry information is lost. Analogous results for other systems can be found in [50, 51]. 5.3.2. Higher-order Spectral Analysis The results discussed in this section were published formerly in [33]. The main objective of higher-order spectral analysis is to assess the presence and effect of nonlinearities in the frequency domain. 5.3.2.1. The Main Concepts The auto bispectrum is defined formally as the Fourier transform of the third-order correlation function of the time series. The discrete bispectrum, appropriate for sampled data, is B(f1 , f2 ) = E[Af1 Af2 A∗f1 +f2 ],
(5.9)
where Afi is the complex Fourier component of the time series at frequency fi , asterisk is complex conjugation, and E[·] is the expected-value, or average, operator. If the three Fourier components in the triple product on the right hand side of Eq. (5.9) are independent of each other (e.g., if they have random phase relationships such as a time series with Gaussian statistics), the bispectrum is zero. It is convenient to recast the bispectrum into its normalized (by the power at each of the three frequencies in the triad) magnitude, called the squared bicoherence, b2 (f1 , f2 ), which represents the fraction of the power of the triad of Fourier components (f1 , f2 , f1 + f2 ) that is owing to quadratic coupling. Similar to the bispectrum, the auto trispectrum is defined formally as the Fourier transform of the fourth-order correlation, and the discrete trispectrum is T (f1 , f2 , f3 ) = E[Af1 Af2 Af3 A∗f1 +f2 +f3 ].
(5.10)
The normalized magnitude of the trispectrum is called the squared tricoherence, t2 (f1 , f2 , f3 ), and is a measure of the fraction of the power of the quartet of Fourier components (f1 , f2 , f3 , f1 +f2 +f3 ) that is owing to cubic nonlinear interactions.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
111
Higher-order spectral analysis can also be used as a means of model validation by comparing bicoherence and tricoherence functions of (measured) data and simulated data obtained by estimated models. 5.3.2.2. The Results Time series of voltage vC1 measured from a realization of Chua’s circuit for the spiral and double-scroll attractors were used to estimate nonlinear polynomial models which can be found in [33]. The circuit parameters used during the experiments were C1 = 11µF, C2 = 45µF, L = 20H, Bp = 1.1 ± 0.2 V, m0 = −0.37 ± 0.04 mS, and m1 = −0.68 ± 0.04 mS. For the spiral attractor (sampled at 12 µs) R = 1.67 kΩ and for the double scroll (sampled at 15 µs) R = 1.64 kΩ. The high inductance (Sec. 5.2.2) reduces the oscillation frequency from O(100 kHz) to O(1 kHz), thus facilitating A/D conversion and recording of the data. Polynomial model parameters were estimated from 8192 values of the voltage across C1 measured after initial transients had decayed. Both the measured and modeled spiral attractors have a relatively narrow power spectrum dominated by a primary peak fp = 2.92 Hz and its harmonics (2fp = 5.85 and 3fp = 8.78 Hz, Fig. 5.10a). The model bicoherence (Fig. 5.10c) indicates quadratic coupling between the primary and its harmonics (e.g., the horizontal band of contours for f2 ≈ 3 Hz, Fig. 5.10c), similar to that observed in the measurements (Fig. 5.10b). Quadratic interactions involving Fourier components with frequencies less than fp are significant in time series from both the data (1 < f2 < 3, f1 ≈ 3 Hz, Fig. 5.10b) and the model (Fig. 5.10c). Tricoherence spectra for the measured time series (not shown) are similar to those from the polynomial model. Cubic interactions between quartets of Fourier components are significant for the spiral attractor, suggesting higher-order coupling between motions with frequencies fp and its harmonics. The coupling between the primary and its harmonics, as indicated by the bicoherence (Fig. 5.10b,c) and tricoherence (not shown) spectra, suggest both quadratic and cubic interactions (as occur in the spiral attractor) are reproduced well by the polynomial model estimated from 8192 values of the voltage measured in Chua’s circuit. The higher-order spectral coherences from the model time series are slightly higher than those from the data, possibly because the model approximates the infinite-order piecewise linear current in the nonlinear resistor with a second- and third-order polynomial, leading to stronger quadratic
October 7, 2008
112
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Fig. 5.10. (a) Power spectra of measured (solid curve) and modeled (dashed curve) voltage vc1 from Chua’s circuit when the system exhibits a spiral attractor. Contours of bicoherence of (b) measured and (c) modeled voltage vc1 . The units of power are arbitrary and the frequencies of the labeled power spectral peaks (panel a) are a) 2.92, b) 5.85, and c) 8.78 Hz. The minimum contour plotted is b = 0.85, with additional contours every 0.1. There are 32 degrees of freedom (from [33]).
and cubic coupling than observed in the actual circuit. Noise in the measured circuit may also reduce higher-order spectral values relative to those from the model time series. 5.3.3. Discrete-time Global Modeling The construction of discrete-time models from data has been thoroughly discussed in the literature. Specifically concerning the double-scroll attractor, [9] and [29] presented the case of NARMAX (nonlinear autoregressive moving average model with exogenous inputs) polynomial and rational models, respectively. 5.3.3.1. Monovariable Polynomial Model The polynomial structure is very attractive because of the simplicity and insight it offers of the systems properties and therefore have been extensively used for reproducing nonlinear dynamics.
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
113
In order to identify dynamically valid models from data generated of Chua’s circuit, the NARMAX model [49] was chosen. The representation of such a model can be as follows y(k) = F ` [ y(k − 1), . . . , y(k − ny ),
u(k − d). . .u(k − d − nu + 1),
(5.11)
e(k). . .e(k − ne )] ,
where ny , nu and ne are the maximum lags considered for the output, input and noise terms, respectively and d is the delay measured in sampling intervals, Ts . Moreover, u(k) and y(k) are respectively the input and output signals. e(k) accounts for uncertainties, possible noise, unmodeled dynamics, etc. and F ` [·] is some nonlinear function of y(k), u(k) and e(k). The function F ` [·] can, of course, be a polynomial-type function with nonlinearity degree ` ∈ ZZ+ . In such a case, to estimate the parameters of this map, Eq. (5.11) should be expressed in prediction error form as ˆ + ξ(k) , y(k) = ψ > (k − 1)θ
(5.12)
where ψ(k − 1) is the regressor vector which contains linear and nonlinear combinations of output, input and noise terms up to and including time k − 1. The parameters corresponding to each regressor are the elements of the ˆ Finally, ξ(k) are the residuals which are defined as the difference vector θ. between the measured data y(k) and the one-step-ahead prediction ψ > (k − ˆ The parameter vector θ can be estimated by orthogonal least-squares 1)θ. techniques [76]. One of the many advantages of such algorithms is that the Error Reduction Ratio (ERR) can be easily obtained as a by-product [17, 48]. In the particular case of Chua’s circuit, one thousand data points of the output y(k), the voltage over capacitor C1 , were used in the identification. The following polynomial model was obtained: y(k) = 3.5230y(k−1)−4.287y(k−2)−0.2588y(k−4)−1.7784y(k−1) 3 +2.0652y(k−3)+6.1761y(k−1)2 y(k−2)+0.1623y(k−1)y(k−2)y(k−4) −2.7381y(k−1)2 y(k−3)−5.5369y(k−1)y(k−2)2 +0.1031y(k−2)3 +0.4623y(k−4)3 −0.5247y(k−2)2 y(k−4)−1.8965y(k−1)y(k−3)2 +5.4255y(k−1)y(k−2)y(k−3)+0.7258y(k−2)y(k−4)2 ˆ −1.7684y(k−4)2 y(k−3)+1.1800y(k−4)y(k−3)2 + ψ > ξ (k−1)θ ξ +ξ(k) ,
(5.13)
October 7, 2008
15:34
114
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
ˆ where ψ > ξ (k − 1)θ ξ is the noise model composed of 20 linear terms of the form ξ(k − j) used to avoid bias. This part of the model was not used in the simulations shown below. The identification of model Eq. (5.13) and the cluster analysis of the double-scroll and spiral attractors have been discussed in detail in [9]. Figure 5.11 show the chaotic double-scroll attractor reconstructed directly from the original data whereas Fig. 5.12 shows the chaotic doublescroll attractor reconstructed using polynomial model Eq. (5.13). For comparison purposes the analysis of the results will be postponed to next section. 5.3.3.2. Monovariable Rational Model Another possibility for the function F ` [·] in Eq. (5.11) is a rational model defined as a ratio of two polynomials [16] y(k) =
a(y(k−1), . . . , y(k−ny ), u(k−1), . . . , u(k−nu ), e(k−1), . . . , e(k−ne )) + e(k), b(y(k−1), . . . , y(k−ny ), u(k−1), . . . , u(k−nu ), e(k−1), . . . , e(k−ne ))
(5.14) where u(k) and y(k) are as before, ny , nu and ne are the maximum lags of the output, input and noise, respectively. Moreover, such lags need not be the same in the numerator and denominator. a(k − 1) and b(k − 1) are polynomial functions nonlinear in the regressors taken up to time k − 1. It is convenient to define the numerator and denominator polynomials in Eq. (5.14) respectively as [16] a(k − 1) =
Nn X
pnj θnj = ψ > n (k − 1)θ n ,
(5.15)
b(k − 1) =
Nd X
pdj θdj = ψ > d (k − 1)θ d ,
(5.16)
j=1
j=1
where θnj , θdj are the parameters of the regressors (of the numerator, pnj , and denominator, pdj ,) up to time k − 1. Nn + Nd is the total number of parameters to be estimated. The use of Eq. (5.14) to perform parameter estimation is not straightforward because such a function is nonlinear in the unknown parameters. An alternative solution to this problem is to multiply both sides of Eq. (5.14)
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
115
Chaos Control for Chua’s Circuits
by b(k − 1) and rearranging terms in order to yield [75] y ∗ (k) = a(k − 1) − y(k) =
Nn X j=1
Nd X j=2
pnj θnj − y(k)
pdj θdj + b(k − 1)e(k)
Nd X
pdj θdj + ζ(k)
j=2
> = ψ> n (k − 1)θ n − ψ d1 (k − 1)θ d + ζ(k) ,
(5.17)
> where ψ > d (k − 1) = [pd1 ψ d1 (k − 1)], θd1 = 1 and
y ∗ (k) = y(k)pd1 =
a(k − 1) pd1 + pd1 e(k) , b(k − 1)
ζ(k) = b(k − 1)e(k) =
Nd X j=1
(5.18)
pdj θdj e(k) ,
(5.19)
where e(k) is white noise. Because e(k) is independent of b(k − 1) and has zero mean, it can be written E[ζ(k)] = E[b(k − 1)]E[e(k)] = 0 .
(5.20)
Equation (5.18) reveals that all the terms of the form y(k)ψ > d (k − 1), because of y(k), implicitly include the noise e(k) which is correlated with ζ(k). This, of course, results in parameter bias even if the noise e(k) is white. The aforementioned correlation occurs as a consequence of multiplying Eq. (5.14) by b(k − 1) and should be interpreted as the price paid for turning a function which is nonlinear in the parameters to one which is linear in the parameters. Algorithms with intelligent structure selection can be found in [18] and [29], to mention a few. Using the same data as described above and also term clustering information [8, 9], the following rational model was obtained [29] y(k) =
1 × 2.5568y(k−1)−1.7594y(k−2)+0.2696y(k−5)+0.6192y(k−1) 3 D −1.0219y(k−2)3 −3.2455y(k−1)2y(k−5)+0.0735y(k−3)3 +0.3444y(k−1)y(k−5)2 −0.4401y(k−2)y(k−5)2
+3.4624y(k−1)y(k−2)y(k−5)+0.1986y(k−1)2y(k−2) +
10 X i=1
θˆi ξ(k − i) +
5 X j=1
θˆj ξ(k − j)2 + ξ(k) ,
(5.21)
October 7, 2008
15:34
116
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
where D = 1+1.5164y(k−1)y(k−2)+.5657y(k−3)y(k−5)−12.8527y(k−2) 2 +0.1948y(k−1)2 +1.7662y(k−2)y(k−5)−0.1409y(k−5)2 −2.0470y(k−1)y(k−5) .
(5.22)
Figure 5.12 shows the chaotic double-scroll attractors reconstructed using polynomial model Eq. (5.13) and rational model Eq. (5.21). Table 5.4 shows the estimated fixed points and largest Lyapunov exponents for both the original system and the identified models. The results suggest that both polynomial and rational models do reproduce the characteristic dynamical features of the original attractor. Such results also seem to indicate that the polynomial representation is slightly better as far as the modeling of attractors produced by Chua’s circuit is concerned. This should come as no surprise because, in fact, the set of equations which describe Chua’s circuit can be expressed in polynomial form. Table 5.4.
Dynamical invariants of data and identified models for Chua’s Circuit
(0,2.24,-2.24)
(0,2.24,-2.24)
Rational Eq. (5.21) – Eq. (5.22) (0,2.37,-2.37)
1.3516 ± 0.0343
1.3350 ±0.0563
1.4218 ±0.0596
Polynomial Eq. (5.13)
Data Fixed Points Largest Lyapunov Exponent
4
3
2
Vc1(k−4)
1
0
−1
−2
−3
−4 −4
−3
−2
−1
0 Vc1(k)
1
2
3
4
Fig. 5.11. Bi-dimensional reconstruction (2000 points) of Chua’s double-scroll attractor. A window of one thousand points of these measured data were used in the identification.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
117
4
4
3
3
2
2
1
1 Vc1(k−4)
Vc1(k−4)
Chaos Control for Chua’s Circuits
0
0
−1
−1
−2
−2
−3
−3
−4 −4
−3
−2
−1
0 Vc1(k)
1
2
3
4
−4 −4
−3
(a)
−2
−1
0 Vc1(k)
1
2
3
4
(b)
Fig. 5.12. Double-scroll attractors reconstructed from (a) simulation of the polynomial model (Eq. (5.13)) and (b) simulation of the rational model (Eq. (5.21)). The x-axis is y(k) and the y-axis is y(k − 4).
5.3.4. Continuous-time Global Modeling The results of building continuous-time models from discrete-time data obtained from Chua’s circuit is briefly described in this section. These results have first appeared in [6]. 5.3.4.1. The Parsimonious Models A 38 term model was obtained from the training time series without any structure selection, therefore, hereafter, such a model will be referred to as the full model. Conversely, the models obtained implementing the structure selection procedure described in [6] and references therein will be referred to as parsimonious models in the sense that all such models will have less than 38 terms. The parsimonious models were obtained by means of structure selection techniques. 5.3.4.2. Topological Characterization The chaotic attractor is embedded in a space spanned by the successive derivatives of the experimentally recorded time series, x. Such an embedding will permit the comparison of the state portrait reconstructed from the experimental data with the portrait generated by the model. The phase space reconstructed from the recorded time series has an embedding dimension equal to 3, which coincides with the dimension of the original phase space. However, five derivative coordinates were used to obtain a global model.
October 7, 2008
15:34
118
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Since the original dynamics can be embedded in a 3D space, only the first three coordinates will be used to characterize the topological structure of the reconstructed state space. Therefore, the first three coordinates are X1 = x(t) x = X2 = x˙ (5.23) X = x ¨ . 3 The chaotic attractor is displayed in Fig. 5.13. Since the system presents an order two inversion symmetry, every point in the time series may have a counterpart on the opposite side of the attractor. Consequently, the reconstructed attractor should also present an inversion symmetry [53]. This information will be used at the end of this section and will prove to be useful in enhancing model quality. The analysis of the dynamics starts with the computation of a firstreturn map obtained from a Poincar´e section. In the case of a symmetric attractor, a fundamental map is used. This can be viewed as a first-return map to a Poincar´e set defined as
(X2 , X3 ) ∈ R2 | X1 = 1.9, X2 < 0 ∪ (X2 , X3 ) ∈ R2 | X1 = −1.9, X2 > 0 , (5.24)
built on the absolute value of one of the two variables (X1 , X2 ). The firstreturn map shown in Fig. 5.14 presents two critical points. The fact that this map does not exhibit any layered structure, confirms the hypothesis of symmetry. 24.0
14.0
Y
4.0
-6.0
-16.0
-26.0 -3.0
-2.0
-1.0
0.0 X
1.0
2.0
3.0
Fig. 5.13. Plane projection of the x-induced attractor generated by the experiments. X = x and Y = x. ˙
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
119
Chaos Control for Chua’s Circuits 20.0
chap5
C!
17.5
1 0 15.0 Yn+1
C2
12.5
2 10.0
7.5
7.5
10.0
12.5
15.0
17.5
20.0
Yn
Fig. 5.14. Fundamental first-return map of the attractor reconstructed from the experimental data.
For details on this procedure, please refer to [52–54]. 5.3.4.3. Model Validation Figure 5.15 shows the projections of the attractors of some estimated model. A further step was to compare the fundamental first-return map computed for attractors generated by the models with the one associated with the experimental data, as shown in Fig. 5.16. Models with 36 and 38 terms are characterized by fundamental first-return maps that have a layered structure which is not observed in the experimental data. The layered structure of the first return maps of models with 36 and 38 terms could be an indication of overparametrization [3]. As a result of this, the system becomes unsymmetrical [54]. On the other hand, the first-return maps of models with 29 and 32 terms are much closer to the experimental one and despite the differences, it seems fair to conclude that structure selection has been able to reduce the complexity of the full model resulting in parsimonious models with improved dynamics. In order to help the algorithm account for the order two inversion, a new model was obtained using a measure which respects the symmetry of the attractor, that is for each point in the differential embedding space, a symmetrical counterpart was included in the data used to estimate the model. For instance, if the original data set had the point x = [X1 X2 . . . X5 ]T , then the mirror image of such a point, −x, was inserted in the data set
October 7, 2008
120
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Experimental
Np=18
Np=29
Np=32
Np=36
Np=38
Fig. 5.15. Plane (x × x) ˙ ≡ (X1 × X2 ) projections of the original attractor, the four parsimonious models and the full model (Np = 38).
before choosing the centers. An Np = 20 term model was found. The first return map of which is shown in Fig. 5.17. It should be mentioned that this model possesses an important property of the experimental flow that is not shared by the other models since all the models mentioned in Fig. 5.16 have symmetric fixed points but the related flow is not symmetric due to the presence of quadratic terms of the form Xi Xj (∀i, j 6= 1). Although improvement in some sense can be observed, the identified model still reveals difficulties. The first-return map does not present the layered (ghostly) structure as before. This is a signature of perfect symmetry as pointed out in [54]. Secondly, the maximum of the first return map shown in Fig. 5.17 does not coincide with the x = y line as observed in other models (Fig. 5.16), this has important consequences in the resulting population of periodic orbits. As a matter of fact, for the Np = 20 model the location of this maximum in relation to the x = y line is very similar to that of the experimental dynamics (Fig. 5.16). Because of the third branch which is increasing, rather than decreasing, as seen in the experimental map (Fig. 5.16), the topology of part of the attractor is different. The linking numbers, that is, the half sum of oriented crossings are computed, reveal some differences between the model and the dynamics underlying the ex-
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
121
Chaos Control for Chua’s Circuits
perimental data. For instance, the pair of periodic orbits encoded by (200) and (10) are displayed for the experimental dynamics (Fig. 5.18(a)) and for the 20 term model (Fig. 5.18(b)). The linking number lk(200, 10) is equal to -2 in both cases but it is noticed that the period-2 orbit of the model is a symmetric orbit, that is, it visits the two scrolls in the same way, whereas it is asymmetric and confined in to a single scroll for the dynamics underlying the experimental data. Such an example confirms that the experimental and the 20 term model attractors are not topologically equivalent although both attractors are rather difficult to distinguish by simple visual inspection. A likely reason for such a difference is the third increasing branch in the first-return map of the model, as shown in Fig. 5.17(b), which is not observed for the experimental dynamics. Interestingly, however, the first-return map of a Chua-type circuit with a smooth nonlinearity [44] also presents a similar increasing branch. Experimental
Np=18
Np=29
Np=32
Np=36
Np=38
Fig. 5.16.
First-return maps for the five different models.
5.3.4.4. Multivariable Polynomial Modeling The following example, first presented in [10], considers data collected from PCChua, with the frequency of the main spectral peak around 1.6 Hz [73]. This enabled collecting the data with a sampling time of Ts = 30 ms.
October 7, 2008
15:34
122
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
0.65
0.75
0.6 0.50 0.55 0.25
0
0.5
Yn+1
Y
1 0.00
0.45 0.4
−0.25
2 0.35
−0.50 0.3 −0.75 −3.5
−2.5
−1.5
−0.5
0.5
X
(a)
1.5
2.5
3.5
0.25 0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Yn
(b)
Fig. 5.17. Phase portrait (a) and the corresponding first-return map (b) generated by the symmetrical model. This model has twenty terms, Np = 20.
Only 1000 observations of each state variable were used to build the model. The data used to build the model are shown in Fig. 5.19. The automatic structure selection scheme followed by an orthogonal extended least squares estimation routine yielded the following model x(k) = 1 .2 103 6 2 03 2 06 x(k − 1 ) + 0.9 7 2 04 7 1 8 3 7 3 3 ×1 0+2 z(k − 1 ) + 0.5 9 8 2 2 1 2 8 3 6 7 5 y(k − 1 ) −0.1 7 1 3 1 5 9 2 4 8 5 8 ×1 0−1 x3 (k − 1 ) + 0.3 9 8 09 7 7 03 6 7 5 ×1 0+5 y(k − 1 )z 2 (k − 1 ) + 0.8 1 9 5 9 3 3 5 4 8 5 6 ×1 0+4 x(k − 1 )z 2 (k − 1 ) −0.1 1 1 07 2 5 6 3 5 09 ×1 0+3 y 2 (k − 1 )z(k − 1 ) y(k) = 0 .8 9 5 2 5 2 009 8 6 6 y(k − 1 ) − 0.2 2 3 3 03 6 8 1 1 1 9 ×1 0+6 z 3 (k − 1 ) + 0.08 3 7 1 4 3 4 9 2 9 4 7 x(k − 1 ) + 0.1 2 9 3 2 5 1 8 03 3 3 ×1 0+3 z(k − 1 ) − 0.6 7 7 2 1 4 8 1 7 4 3 2 ×1 0−3 x3 (k − 1 ) + 0.1 8 1 08 5 2 6 8 7 8 3 ×1 0−3 x2 (k − 1 ) −0.2 04 05 04 4 6 8 1 2 ×1 0+1 x(k − 1 )y(k − 1 )z(k − 1 ) z(k) = 0 .9 3 8 2 6 5 05 7 18 6 z(k − 1 ) − 0.6 3 7 4 8 5 6 8 4 2 2 0×1 0−3 y(k − 1 ) −0.3 6 2 1 3 4 02 01 8 5 ×1 0−4 x(k − 1 ) + 0.3 1 5 3 6 3 09 3 5 7 5 ×1 0−5 x2 (k − 1 ) + 0.6 3 2 8 6 8 3 4 5 4 4 1 ×1 0−5 y 2 (k − 1 ) −0.6 4 3 3 1 8 6 8 2 3 1 2 ×1 0−5 x(k − 1 )y(k − 1 ) + 0.1 8 7 6 1 9 3 2 1 5 3 7 ×1 0−2 x(k − 1 )z(k − 1 ).
(5.2 5)
As before (Sec. 5.3.3.1), only the deterministic part of the model is shown. It is important to notice that model Eq. (5.25) cannot produce a mathematically symmetrical attractor because it includes even parity terms [54]. When symmetry conditions were imposed during modeling, the new model did not settle unto the double scroll attractor and state estimation results did not improve. Besides the polynomial model Eq. (5.25) multilayer perceptron (MLP)
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
123
Chaos Control for Chua’s Circuits
25.0
15.0
y
5.0
-5.0
-15.0
(10) (200) negative crossings
-25.0 -0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
(a) 0.75
0.50
Y
0.25
0.00
−0.25
−0.50
(10) (200) negative crossings
−0.75 0.0
0.5
1.0
1.5
2.0
2.5
3.0
X
(b) Fig. 5.18. (a) Experimental dynamics, and (b) model dynamics. The linking numbers between periodic orbits encoded by (200) and (10) is equal to -2 for both the experimental dynamics and the 20 term model dynamics. Nevertheless, the period-2 orbit does not present the same symmetry properties. See text for a possible explanation.
neural networks were trained from the same set of data. Three networks were trained, one for each state variable. The input layer in each case was the same, namely [x(k − 1), y(k − 1), z(k − 1)]> . The activation functions of the nodes in the single hidden layer were hyperbolic tangent and the single output node was linear in all three networks. In this section we discuss the results obtained by two set of networks. The first set was composed of three fully connected networks with two nodes in the hidden layer and the second was composed of three fully connected networks with seven nodes in the hidden layer. Such neural models will be referred to as MLPs2 and
October 7, 2008
15:34
124
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
−3
x 10 5 4 3
z(k) (mA)
2 1 0 −1 −2 −3 −4 −5 1 0.5 0 −0.5
y(k) (V)
−1
−5
−4
−3
−2
−1
0
1
2
3
4
5
x(k) (V)
Fig. 5.19. The measured double scroll attractor of the implemented electronic oscillator. There are the data used to build model Eq. (5.25).
MLPs7, respectively. Briefly, MLPs2 also settled to limit cycle and MLPs7 showed a very long chaotic transient— consistent with the double-scroll attractor— before settling to a spiral strange attractor. It was believed that all the estimated models— Eq. (5.25), MLPs2 and MLPs7— could be competitive in the implementation of a Unscented Kalman Filter – UKF [40]. However, it was desired to verify if models with improved global dynamics would outperform locally optimal models. In order to investigate this issue slight changes were implemented in model Eq. (5.25) a posteriori, as described in the following paragraph. The free-run of model Eq. (5.25) settles to the attractor shown in Fig. 5.20(a). Such an attractor seems to be close to a genuine solution of the system in the sense that it could be the “stable version” of one of the unstable periodic orbits that compose the original attractor. When this is the case, it has been argued that the model can be perturbed (this can be done in different ways) in order to become chaotic [1]. In the present case this is achieved by slightly increasing the underlined parameter. For instance, if 0.92427441 is used, model Eq. (5.25) settles to the double-scroll attractor shown in Fig. 5.20(b)a . In what follows we shall refer to this as the perturbed double-scroll model.
a Other
parameter values were tested for perturbing model Eq. (5.25) in order to obtain the double-scroll attractor. Nevertheless, the greater the disturbance (i.e. the farther from the optimal set of parameters) the worse the performance of the UKF in estimating the states, although the model might display improved global features.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
125
Chaos Control for Chua’s Circuits
−3
x 10 5 4 3 2 1
z(k)
0 −1 −2 −3 −4 −5 1 0.5 0 −0.5 −1
y(k)
−5
−4
−3
−2
−1
2
3
5
1
4
0
2
3
5
1
4
0
x(k)
(a) −3
x 10 5 4 3 2 1
z(k)
0 −1 −2 −3 −4 −5 1 0.5 0 −0.5
y(k)
−1
−5
−4
−3
−2
−1
x(k)
(b) Fig. 5.20. Attractors produced by identified model (a) without and (b) with parameter perturbation, with initial conditions x(0) = 3.2, y(0) = 0.37 and −0.009. From [10].
5.3.5. State Estimation The results in this section were first presented in [10]. The UKF with the identified model Eq. (5.25) was used to estimate the full state vector of the electronic circuit using only the x component to drive the filter. The following situations were tested: i) using Eq. (5.25) as obtained from the modeling step; ii) slightly perturbing the underlined parameter of Eq. (5.25) in order to approximate the resulting attractor to the original double scroll attractor; iii) using Eq. (5.25) but jointly estimating the three parameters in italic (which correspond to the terms of each equation with higher ERR index), iv) the same as in (i) but using a null process noise covariance
October 7, 2008
15:34
126
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
matrix Q. It is important to declare that in the aforementioned situations, except in (iv), Q was taken as a diagonal matrix with elements equal to the onestep-ahead prediction error variance of the respective modelb . In case (iii), where three parameters were also estimated, the elements of Qθ were taken as the corresponding parameter variances obtained as a byproduct from the extended least squares estimator used during modeling step. The results are summarized in Table 5.5. Table 5.5. Normalized root mean square (RMS) error of estimated states for the electronic system. The UKF was implemented with the identified model Eq. (5.25) — in several contexts (see text)— and with the neural models MLPs2 and MLPs7. From [10]. situation investigated x y z (i) model 2.80×10−7 % 2.58% 0.87% (ii) perturbed model 3.75×10−7 % 5.85% 2.86% 3.70×10−7 % 3.03% 1.24% (iii) joint parameter estimation (iv) model with Q = 0 88.46% 85.46% 76.46% 2.57×10−7 % 2.06% 1.18% (v) MLPs2 9.33×10−7 % 1.27% 0.86% (vi) MLPs7
The very low values of RMS for the estimation errors of the x component are a consequence of low measurement noise in PCChua experimental setup (Sec. 5.2.3) and due to the fact that the x variable is used to drive the filter. An interesting remark that can be made based on the results of Table 5.5 is that although the perturbed model is, in a sense, closer to the original system (for instance, both have a positive Lyapunov exponent), the unperturbed model— which has a lower one-step-ahead prediction error— is better suited for use within the UKF. When the UKF was implemented using Eq. (5.25) then, quite naturally, the UKF estimates also settled to the same attractor (possibly just a limit cycle) shown in Fig. 5.20a. This explains the very high values of RMS for situation (iv) shown in Table 5.5. This problem can be circumvented by setting a lower bound for the covariance matrix P by properly setting the process noise covariance matrix Q. Otherwise, when convergence was achieved, the trace of matrix P would tend to zero. In so doing the filter remains more active and the most recent measurements gain more weight compared to the model and as a result the UKF settles to a doublescroll attractor, see Fig. 5.21(a). In other words, there are two important b Obtained
by one-step-ahead simulation using each model.
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
127
sources of information in the filter: on the one hand, there is the model that propagates the sigma points and, on the other hand, there are the measurements. Suitably defining Q and R is a way of weighting differently these two sources. So as to express lack of confidence in the model, the trace of Q should be increased. Similarly, the trace of R ought to be directly related to the degree of uncertainty in the measurements. When parameters are estimated in addition to the states, the estimated parameters converge to the values obtained during the modeling stage as indicated in Fig. 5.22. Consequently situations (i) and (iii) have close performance indices in Table 5.5. Table 5.6 summarizes the UKF performance when extra white Gaussian noise is added to the x component in the case of polynomial and neural models. As can be seen, the UKF implemented with the neural model with 7 hidden nodes is significantly more robust to noise than the filters implemented with the other models. At first sight this could seem inconsistent with the results shown in Table 5.5 where MLPs7 has the highest error for the x variable. It should be noticed however that this weakness of MLPs7 does not appear in Table 5.6 because the filters are driven by the x variable, that is, poor performance in predicting the driving signal is not as serious as poor performance in predicting the remainder of the state vector, in which case the predictions are all we have. Table 5.6. Normalized root mean square (RMS) error of estimated states for the electronic oscillator. The UKF was implemented with the identified model Eq. (5.25) and with the neural models MLPs2 and MLPs7. The x-component, that drives the filter, was corrupted with extra additive white Gaussian noise. model/noise x (%) y (%) z (%) Polyn 10% 1.97 3.37 1.42 MLPs2 10% 3.48 5.63 2.55 MLPs7 10% 1.01 2.88 1.37 Polyn 25% 4.35 6.08 3.08 MLPs2 25% 8.20 12.66 5.54 MLPs7 50% 1.57 3.53 1.84 Polyn 50% 17.82 17.93 12.84 MLPs2 50% 25.56 27.20 18.13 MLPs7 50% 2.95 5.09 2.75
October 7, 2008
128
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
−3
x 10 5 4 3 2
z(k)
1 0 −1 −2 −3 −4 −5 1 0.5 0 −0.5 −1
y(k)
−5
−4
−3
−1
−2
0
1
2
3
4
5
x(k)
(a) −3
5
x 10
4 3
z(t), z(k) (mA)
2 1 0 −1 −2 −3 −4 −5
0
5
10
15
t = kTs (s)
(b) Fig. 5.21. The estimated double scroll attractor of the implemented electronic oscillator. Estimation was performed by the UKF with model Eq. (5.25) and the x component driving the filter. (a) Full state vector, (b) detail of the z component (-·-) measured data, and (—) estimated component zˆ(k). In the plot (a) the transient at the beginning was also included to give an idea of how quickly the filter settles. From [10].
5.4. Chua’s Circuit Control and Synchronization The control (and synchronization) of Chua’s circuit has been chosen as a benchmark in nonlinear control of dynamical systems by many researchers around the world. This is explained by the simultaneous conjugation, on Chua’s circuit, of complex behavior, simple construction, and manageable mathematical
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
129
Chaos Control for Chua’s Circuits
(a) 1.4
θ
1
1.3 1.2 1.1 1
0
5
10
15
20
25
30
20
25
30
20
25
30
(b)
θ
2
1 0.8 0.6 0.4
0
5
10
15
(c)
θ
3
1.2 1
0.8 0
5
10
15
t = kT (s) s
Fig. 5.22. The solid lines (—) show the results obtained from the joint estimation of the three parameters in italic p in Eq. (5.25). The dotted lines (·) indicate plus and minus one standard deviation (± diag{Pθ }) of the parameters estimated by joint UKF. The dashed lines (- -) indicate the values for these parameters estimated by extended least squares during the modeling step. From [10].
description composed by a set of only three piecewise affine differential equations. As a consequence, to control and/or to synchronize Chua’ circuits has become a paradigm in nonlinear control systems theory. As a sample of modern nonlinear control strategies used to control Chua circuits, or one of the many variations of this famous chaotic oscillator, it is possible to register: feedback linearization [55], backstepping [74], sliding mode [39], and fuzzy control [23], among others. In this section, a sample of results from the application of different control and synchronization strategies to the circuit of Chua will be presented. The aim is to show how Chua’s circuit can be employed to draw deep conclusions on the more general problem of controlling nonlinear systems.
October 7, 2008
130
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
5.4.1. General Considerations – Control Objective, Available Energy and Mathematical Modelling The control of nonlinear dynamical systems is undoubtedly a vast field of research, which has been approached from many different points of view (see [43, 64, 66] and references therein). The lack of a universal theory, as powerful as the well known Linear System Theory [27], with which one would be capable of explaining the plethora of phenomena found in these systems, almost prevents the achievement of general results. Despite this fact, in [12] it was possible to elaborate broader conclusions on the performance of controllers for nonlinear dynamics, by taking into account a number of aspects such as: (i) maximum control effort allowed (energy limits); (ii) the dissimilarities of the control objective (goal dynamics) when compared to the natural behavior of the uncontrolled system; and (iii) the controller structure complexity itself; i.e. the necessity or not of accurate models of the system to be controlled in the design of the control law. The tradeoff among these aspects can be clearly illustrated on the control and synchronization of Chua circuits. For example, when the control objective is to force one or more state variables of the chaotic oscillator to follow an arbitrary reference; e.g. a sinusoidal signal; one expects that complex and/or high energy controllers would be necessary, as it will be presented in Sec. 5.4.4. On the other hand, if the reference signal is constant at a specific value such that the problem becomes one of stabilization of unstable fixed points of Chua’s circuit, the demanded energy could be reduced, and the controller could be made simpler than the one used in the former case, as it will be seen in Sec. 5.4.2. Another example is the stabilization of unstable periodic orbits (UPO); which are the fundamental bricks in the system chaotic attractor, and therefore are close to the natural behavior of the uncontrolled system. In this case, one expects that simple and low energy controllers would suffice. This is indeed verified in the famous OGY chaos control method [61], whose variations have been applied to Chua’s circuit in [23] and [34]. Similarly, although not obvious, in the problem of synchronization of unidirectionally coupled identical nonlinear oscillators, if one considers that this is indeed a control problem where the reference signal is the trajectory produced by the so-called master system, such that a slave system has to be controlled in order to follow it, simple and/or low energy controllers could be employed. This is a consequence of the fact that the control
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
131
Chaos Control for Chua’s Circuits
objective is to force the slave system follow one of its natural trajectories after synchronization has been attained. If the goal trajectory belongs to the library of natural behaviors of the slave then, after synchronization has been reached, the control effort should vanish in the absence of noise and perturbations. This will be discussed in the next section. 5.4.1.1. The ITVC Principle Interesting considerations naturally arise from the close relationship between control and synchronization of nonlinear oscillators. In [72] for continuous-time systems, and in [69] for discrete-time systems, a general principle called Information Transmission Via Control – ITVC was presented and proved, which relates the control effort u(t), demanded to keep the systems synchronized, to the perturbation (or information) p(t) signal injected at the master oscillator, as shown in Fig. 5.23.
p(t)
Master x
sx (t) η (t) Controller
u(t)
Slave y
sy (t)
Fig. 5.23. Unidirectionally coupled systems or master-slave configuration. p(t) and u(t) are the perturbation and the control signals, respectively; x(t) and y(t) are the state vectors of the master and the slave nonlinear oscillators, which tend to synchronize by means of the control action u(t). sx (t) and sy (t) are measured signals from the oscillators. The Information Transmission Via Control principle [72] states that u(t) → p(t), in the absence of noise η(t), as long as y(t) → x(t).
Consider the following information transmission system based on additively perturbed nonlinear oscillators: x˙ = f (x) + p(t), y˙ = fˆ(y) + u(t),
(5.26)
sx = h(x) + η(t),
(5.28)
(5.27)
where x ∈ Rn is the master oscillator state vector, f : Rn → Rn , p(t) = [p1 (t) p2 (t) . . . pn (t)] is a vector of information signals, where it is allowed
October 7, 2008
15:34
132
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
to have pk (t) = 0 for some values of k ∈ {1, 2, . . . , n}; sx ∈ Rq is the vector of transmitted signals, such that q ≤ n; h : Rn → Rq is a general mapping; η(t) = [η1 (t) η2 (t) . . . ηq (t)] is a vector of noise signals; y ∈ Rn is the slave oscillator state vector; fˆ : Rn → Rn , and u(t) = [u1 (t) u2 (t) . . . un (t)] is a vector of control signals, also represented as continuous functions of time. In this case, the following can be established: Theorem 5.1. (ITVC – Information Transmission Via Control Principle). Define the synchronization error vector as e = x − y.
(5.29)
Suppose that the systems Eq. (5.26) and Eq. (5.27) have C 0 (continuous) ˆ ˆ vector fields f (x) and f(y), respectively. In addition, f (x) = f(x), ∀x ∈ Rn , i.e. the two autonomous oscillators are identical. Under these assumptions, one has the following result: lim kek = 0 ⇒ lim kp(t) − u(t)k = 0,
t→∞
t→∞
(5.30)
where k · k denotes the Euclidean norm. An important fact, concerning the general result of the ITVC principle, was also proved in [72]. In practice, identical synchronization (kek → 0) is not possible due to unavoidable noise and parameters mismatch, such that the synchronization error does not go to zero, but instead it remains limited: 0 ≤ kek ≤ E < ∞. When this occurs, high-energy controllers should not be used to recover p(t), as can be deduced from the following result: Theorem 5.2. (Information recovery when both systems are not identically synchronized). Suppose that f : Rn → Rn is a C ω (analytic) function representing the vector fields of both master Eq. (5.26) and slave Eq. (5.27) ˆ oscillators (f (x) = f(x), ∀x ∈ Rn ). In addition, suppose that 0 ≤ kek ≤ E < ∞. Considering the recovering error defined as ed = p(t) − u(t), then ked (t)k → kek, ˙ as long as E → 0. More specifically, it is possible to write that ked (t)k ≤ ke(t)k ˙ + ρ(y)E + φ(E 2 ) ked (t)k ≥ ke(t)k ˙ − ρ(y)E − φ(E 2 )
(5.31)
(5.32)
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
133
where ρ(y) = kDfy k is the norm of the Jacobian matrix Dfy of f (e + y) evaluated around e = e0 = 0, which depends on the value of y. The function φ(E 2 ) : R → R+ represents high order terms in E that vanish when E → 0, i.e. limE→0 φ(E 2 )/E = 0. By applying the above theorem, one sees that the error ed (t) in recovering the perturbation (information) approaches the derivative of the synchronization error e. Therefore, the theorem clearly establishes that it is possible to achieve a dynamic condition arbitrarily close to the identical synchronization (kek ≈ 0), and still do not recover accurately the original perturbation signal p(t), if the controller structure is such that it results in kek ˙ 0. Such controllers; i.e. the control laws that lead to the condition
h
i
kek ˙ = f (x) − fˆ(y) + [p(t) − u(t)] 0;
are called in this context “high-energy controllers”, because usually it is observed negligible mismatch between master and slave oscillators (kf (x) − fˆ(y)k ≈ 0); and the perturbation signal is generally kept small (kp(t)k ≈ kek) in order to not destabilize the master oscillator. Under these assumptions, it follows that kek ˙ 0 ⇒ ku(t)k 0. In Sec. 5.4.4.2 an example of this situation will be discussed. The ITVC principle, as stated in [69], has also been used as the basis for model validation procedures that rely on the control effort spent to synchronize identified discrete-time models with the corresponding acquired data set from a real system [7]. Similar remarks concerning the fact that low-energy controllers should be used in this validation approach (due to the impossibility of achieving perfect synchronization between real data and identified model) were also proved in [69]. 5.4.2. Linear State Feedback In this section, the control and synchronization of Chua’s chaotic oscillator is investigated based on the design of linear state feedback controllers. A method for selecting the controller gain matrix is proposed based on the use of the optimization technique known as the ellipsoid algorithm [31]. The formulation presented here can be viewed as an extension of the work in [70] where the focus was the controller structure design. The objective is to provide a more general method to determine the non zero elements in the controller gain matrix in order to issue stable controlled systems.
October 7, 2008
15:34
134
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
The connection between the control and synchronization problems is explored in detail. It is shown that the synchronization problem is similar to the equilibrium point stabilization problem concerning the linear controller design requirements. Consider the controlled circuit of Chua (Fig. 5.6, page 105) described by the set of differential equations Eq. (5.7), repeated here, but substituting z˜ by z for clarity purposes: x˙ = p [y − x − f (x)] + ux (t, ~x), y˙ = x − y + z + uy (t, ~x), z˙ = −qy + rz + uz (t, ~x),
(5.33)
where p = C2 /C1 and q = −R2 C2 /L. Note that the derivatives were taken relative to the normalized time variable t = t0 /RC2 , where t0 represents the actual time (Sec. 5.2.1). The nonlinear function f (x) = Rid (x) in Eq. (5.33) is proportional to the current through the Chua’s diode, which is given by Eq. (5.2), page 100. The control inputs ux (t), uy (t) and uz (t) in Eq. (5.33), assuming linear state feedback control, can be rewritten in vector form as >
~u(t) = [ux uy uz ] = K(~x − ~xref (t)),
(5.34)
>
where ~x = [x y z] is the state vector, the reference vector is denoted > by ~xref (t) = [xref (t) yref (t) zref (t)] and K ∈ R3×3 is the controller gain matrix. Using the Chua’s diode characteristic Eq. (5.2) together with Eq. (5.34), it is possible to rewrite Eq. (5.33) in matrix form as ~x˙ = Ax ~x + ~bx + K (~x − ~xref )
(5.35)
where Ax =
A1 ∈ R3×3 , if x ∈ [−Bp , +Bp ], A2 ∈ R3×3 , if x < −Bp or x > +Bp ,
> ~b1 = pR[Bp (m0 − m1) 0 0] , if x < −Bp , ~bx = ~b2 = [0 0 0]> , if x ∈ [−Bp , +Bp ], ~b = pR[B (m − m ) 0 0]> , if x > +B . 3 p 1 0 p
(5.36)
(5.37)
Actually, the matrix Ax is the Jacobian matrix of the nonlinear vector field of the autonomous chaotic oscillator. Defining the error vector as ~e = ~x − ~xref ,
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
135
it is possible to derive the following differential equation describing the error dynamics as a linear time-variant non-autonomous system represented by d~e d~xref = Ax (~e + ~xref ) + ~bx + K~e − ; dt dt ~e˙ = Ak ~e + ~η (t), Ak = Ax + K, η~(t) = Ax ~xref (t) + bx − ~x˙ ref (t).
(5.38) (5.39) (5.40)
Assuming that it is possible to adequately choose the gain matrix K in Eq. (5.39) so that the instantaneous eigenvalues of Ak have negative real parts, the time evolution of the error vector inside each linear region is given by [27] ~e(t) = exp [Ak (t − t0 )] ~e(t0 ) + conv [~ η (t) , exp [Ak (t − t0 )]]
(5.41)
where t0 corresponds to the time instant when the system enters the linear region, and conv [· , ·] denotes the linear convolution operator. Note that, despite the complexity in Eq. (5.41), from the above assumption of asymptotic stability, the error behavior can be approximated by h i ~e t |λc |−1 ≈ conv ~η (t) , ~h(t) , (5.42) ~h(t) = exp [Ak (t − t0 )] ,
λc = min |real{λ(Ak )}|, λc < 0,
Moreover, the greater the value of |λc | the faster the decay of the components in the vector function ~h(t), in such a way that the expression Eq. (5.42) becomes similar to h i ~e(t > T ) ≈ conv ~η (t) , ~δ(t) , (5.43) where ~δ(t) is a vector of impulsive functions whose magnitudes are proportional to the areas under the graphs of the ~h(t) components, which in turn become smaller for greater values of |λc |. This qualitative analysis was carried out to stress that the larger the absolute values of the real parts of the instantaneous eigenvalues of Ak in Eq. (5.39), the smaller the maximum steady-state error, as can be seen from Eq. (5.43). This effect can be obtained by means of gain augmentation, e.g. through the use of large values in a diagonal matrix K; which is a well known result from linear control theory.
October 7, 2008
136
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Assuming that the matrix K was appropriately chosen, reducing the error vector norm in order to guarantee that the state vector ~x and the reference state ~xref will be in the same affine region after a time interval T , the Eq. (5.40) can be rewritten as η~ (t > T ) = Axref ~xref (t) + bxref − ~x˙ ref (t).
(5.44)
Note that the first two terms in Eq. (5.44) constitutes the very nonlinear vector field of an autonomous Chua’s circuit whose state vector is ~xref . Suppose that the control objective is to stabilize an equilibrium point p~eq of the chaotic oscillator. In this case, Eq. (5.44) becomes ~xref = p ~eq = [xeq yeq zeq ]> , ⇒ ~η (t > T ) = 0. ~x˙ ref = 0, ~ Axeq p~eq + bxeq = 0,
(5.45)
A similar result can be derived when the reference trajectory is generated by another Chua’s oscillator with the same parameters, i.e. a masterslave synchronization problem, ~x˙ ref = Axref ~xref + ~bxref ⇒ η~(t > T ) = 0.
(5.46)
The above result reveals that there exists a strong similarity between the Chua’s circuit equilibrium point stabilization and synchronization problems. In both cases, the requirement on providing larger absolute values for the instantaneous eigenvalues of Ak in Eq. (5.39) can be softened due to the nature of the control objective. The most important requirement concerning the matrix K selection is to guarantee an asymptotically stable matrix Ak in Eq. (5.38) for all t ≥ 0. Clearly, from Eq. (5.36) and Eq. (5.39), at any instant, only one of two possible characteristic polynomials will be relevant in the stability analysis. Applying the Routh criterion [27] on the coefficients of the characteristic polynomials det [(A1 + K) − λI] = 0 ⇒ λ3 + α2 λ2 + α1 λ + α0 = 0, 3
(5.47)
2
det [(A2 + K) − λI] = 0 ⇒ λ + β2 λ + β1 λ + β0 = 0, where I ∈ R3×3 is the identity matrix and det[·] denotes matrix determi-
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
nant, issues the following set of restrictions α0 g1 : g : α α 1 2 − α0 2 g3 : α2 Ω: g4 : β0 g : β β − β0 5 1 2 g6 : β2
> > > > > >
0, 0, 0, 0, 0, 0.
chap5
137
(5.48)
Note that the coefficients αn and βn , n = 0, 1, 2; are, in general, nonlinear functions of the elements kij ; i, j = 1, 2, 3; in matrix K. The great complexity exhibited in Eq. (5.48) can be reduced using simpler control structures. For example, assuming that ux (t) = uy (t) = 0 in Eq. (5.33) so that, the circuit is controlled only by the voltage source in series with the inductor L shown in Fig. 5.6, page 105, the matrix K becomes 0 0 0 K = 0 0 0 , (5.49) k1 k2 k3
and there will be only two nonlinear restrictions in Eq. (5.48), namely g2 and g5 . Clearly, other control structures can be tested based on the same approach, and the set of restrictions Eq. (5.48) bounds a region Ω in the m-dimensional controller parameter space for which the matrix Ak in Eq. (5.39) is asymptotically stable; where m is the number of non zero elements in the matrix K. Concerning the control problem, an additional requirement is to maximize the minimum absolute eigenvalue of Ak in Eq. (5.39), in order to achieve a better tracking performance. This means that there exists an optimal set of control parameters ~k ∗ inside the region Ω. On the other hand, when dealing with the synchronization problem, the requirement on the minimum absolute eigenvalue of Ak is not critical, as discussed previously on the similarity between the synchronization and the equilibrium point stabilization problems. Considering only the restrictions gi in Eq. (5.50) and using the centercut ellipsoid algorithm described in [31], examining the restrictions in a cyclical order, it is possible to select a viable matrix K that leads to asymptotic stability in the synchronization problem. In the case of the general control problem; i.e. arbitrary reference signal ~xref (t); note that the coefficients α2 and β2 in Eq. (5.47) are the negative sum of the stable eigenvalues of (A1 +K) and (A2 +K), respectively. Hence,
October 7, 2008
138
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
a possible way to maximize the minimum absolute eigenvalue of Ak is to solve the following optimization problem max{α2 + β2 } s. t. gi > 0; i = 1, 2, . . . , 6;
(5.50)
where the restrictions gi were defined in Eq. (5.48). The use of the ellipsoid algorithm on Eq. (5.50), issues a gain matrix K well suited for control purposes. A simulation procedure was carried out using the following set of Chua’s circuit parameters: C1 = 23.5µF, C2 = 235µF, L = 43.43H, R = 1.8kΩ (see Fig. 5.6, page 105); and m0 = −0.36mS, m1 = −0.68mS and Bp = 1.0V as Chua’s diode parameters; Eq. (5.2), page 100. The results are shown in Fig. 5.24 to Fig. 5.26 for the control structure represented by the gain matrix K shown in Eq. (5.49) with k1 = −19.00, k2 = −46.47 and k3 = −3.40. The objective is to demonstrate that a badly designed linear controller, exhibiting large error signal as shown in Fig. 5.24 and Fig. 5.25, can still efficiently synchronize two circuits of Chua as shown in Fig. 5.26. Note that the gain and offset distortions for a reference signal completely inside one affine region of Eq. (5.35) is different from the signal distortion observed for a reference signal inside another affine region, as shown in Fig. 5.24 and Fig. 5.25 respectively. This is a result of different sets of eigenvalues for (A1 + K) and (A2 + K) in Eq. (5.38) that leads to different temporal behaviors for the state vector ~x in each corresponding region described by Eq. (5.35). The above results indicate that the synchronization problem is quite similar to the equilibrium point stabilization problem concerning the stability requirements on the gain matrix controller selection, and that the nature of the control objective is indeed crucial in the controller performance evaluation. 5.4.3. Robust Control and Synchronization via Linear Matrix Inequalities The synchronization problem of two Chua’s circuits via state feedback can also be addressed using optimization techniques based on the so called linear matrix inequalities (LMIs). The main advantage of this kind of strategy is that the control law can be derived in a systematic way if the system model is available. As shown in the previous section, the nonlinear Chua’s circuit can be modelled as a piecewise linear framework. Roughly the idea is that Chua’s
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
139
Chaos Control for Chua’s Circuits
2.5
2
x, x ref (V)
1.5
1
0.5
0
0
2
4
6
8 t (s)
10
12
14
16
10
12
14
16
(a) 0.8
0.6
0.4
0.2
rz (V)
0
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
8 t (s)
(b) Fig. 5.24. Chua’s circuit linear control using only a voltage source in series with inductor L (see Fig. 5.6). Control activated at t = 2.5s. (a) (—) Voltage across capacitor C 1 ; (− −) Reference signal: xref (t) = 1.5+0.5 sin(0.4πt), yref (t) = 0, zref (t) = 0. (b) Applied control signal.
circuit can be represented by a set of “local” systems. With such modelling procedure, diffused techniques from the robust H∞ control theory can be easily applied to these systems. Section 5.4.3.1 presents the LMI based formulation to obtain the control gain to synchronize a master-slave scheme. To evaluate the synchronization control performance, the information transmission problem is addressed in Sec. 5.4.3.2 using the PCChua
October 7, 2008
140
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
2.5
2
1.5
x, x ref (V)
1
0.5
0
−0.5
−1
−1.5
0
2
4
6
8 t (s)
10
12
14
16
10
12
14
16
(a) 1.5
1
0.5
rz (V)
0
−0.5
−1
−1.5
−2
−2.5
0
2
4
6
8 t (s)
(b) Fig. 5.25. Chua’s circuit linear control using only a voltage source in series with inductor L. Control activated at t = 2.5s. (a) (—) Voltage across capacitor C 1 ; (− −) Reference signal: xref (t) = 0.5 sin(0.4πt), yref (t) = 0, zref (t) = 0. (b) Applied control signal.
(Sec. 5.2.3); i.e. the information signal can be recovered in the slave system using a coherent demodulation technique, if the robust synchronization is established. Notice that other robust H∞ synchronization strategies for coupled Chua’s circuits based on LMIs have been reported in the recent literature; e.g. a fuzzy control based approach [60] and a time-delay feedback control [67].
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
141
Chaos Control for Chua’s Circuits
3
2
0
x, x
ref
(V)
1
−1
−2
−3
0
2
4
6
8 t (s)
10
12
14
16
12
14
16
(a) 0.6
0.4
0.2
rz (V)
0
−0.2
−0.4
−0.6
−0.8
0
2
4
6
8 t (s)
10
(b) Fig. 5.26. Chua’s circuit synchronization using only a voltage source in series with inductor L. Control activated at t = 5.0s. (a) (—) Voltage across capacitor C 1 ; (− −) Genuine chaotic trajectory: ~ x˙ ref = Axref ~ xref + ~bxref . (b) Applied control signal.
5.4.3.1. Robust H∞ Synchronization Consider a synchronization scheme of nonlinear discrete time systems with state and output transitions described as: m xm k+1 = A(ρk )xk + B(ρk ) (5.51) Master y m = Cxm k k xsk+1 = A(ρk )xsk + B(ρk ) + uk (5.52) Slave y s = Cxs k k
October 7, 2008
15:34
142
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
n where xm,s ∈ Rn and xm,s k k+1 ∈ R denote the state vectors in the discrete time instants k and k + 1, respectivelyc , ykm,s ∈ Rm denotes the measured output vectors, uk ∈ Rn denotes the synchronization control vector, A(ρk ) ∈ Rn×n represents the dynamic matrix of the system, B(ρk ) ∈ Rn is the affine term, C ∈ Rm×n is the output matrix and ρk ∈ Rp is a parameter vector that depends on time k. The synchronism is achieved when dynamical systems simultaneously describe a common trajectory on the state space. By defining the synchronization error as the difference between the s master and slave states, εk = xm k − xk , and assuming a control law m s n×m uk = L(ρk )(yk − yk ), where L(ρk ) ∈ R is the matrix of synchronization control gains, the synchronism of the master and slave systems can be investigated by checking the synchronization system error given by:
εk+1 = (A(ρk ) − L(ρk )C)εk
(5.53)
Global synchronization of the master and slave systems means that, for any initial state xs0 of the slave, and any initial state xm 0 of the master, the convergence condition presented below is verified lim k εk k→ 0,
k→∞
∀ε0 ∈ Rn
(5.54)
where k · k is the Euclidean norm. The difficulty to establish the general conditions for the synchronization of global chaotic systems has led many researchers to consider particular classes of nonlinear systems. In the literature, a class of systems commonly investigated is the so-called Lur’e systems. Numerous approaches such as those ones given in [22, 38, 68] deal with the synchronization of this class of systems. This section focus on the synchronization of a class of Lur’e discrete time systems represented by local descriptions in the state space. The nonlinear systems are then modelled in a piecewise linear framework with the state SN space partitioned into N regions Ri with i=1 Ri ⊆ Rn . At each discrete time instant k, the state vector xk visits a unique region Ri of the state space associated to a set of constant matrices Ai , Bi e Li . Notice that such a modelling procedure allows to apply some techniques borrowed from the robust control theory.
c ‘m’
and ‘s’ denote the systems master and slave, respectively.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
chap5
143
Consider a scenario where the master system is disturbed by exogenous signals wk ∈ Rq in their state and measured output: Master
m xm k+1 = Ai xk + Bi + Ei wk y m = C xm + D w i k i k k
(5.55)
where Ei ∈ Rn×q and Di ∈ Rm×q are weighting matrices. In this noisy context the synchronization of the master and slave systems could suffer a performance degradation and, in the worst case, the synchronization could be completely lost. The determination of a control law uk that guarantees a robust (insensible to the presence of noise) synchronization of the master and slave systems becomes fundamental. The results stated in the sequel deal with the problem of chaotic system synchronization in a noise environment. The main result is based on the concept of robust H∞ stability of the error system Eq. (5.53) rewritten as: Error
(
εk+1 = (Ai − Li C)εk + (Ei − Li Di )wk ˜ k + Dw ˜ k zk = Cε
(5.56)
where zk ∈ Rr denotes a weighing vector of the synchronization error and ˜ ∈ Rr×q are weighting matrices. C˜ ∈ Rr×n and D The robust H∞ -norm performance index can be guaranteed to obtain the upper bound γ to the induced `2 gain defined as: k z k2 <γ k w k2 kwk2 6=0 sup
(5.57)
where k · k2 represents the `2 -norm. Notice that the induced `2 gain is the rate between the energy of the weighting synchronization error signal ‘z’ and the energy of the exogenous disturbing signal ‘w’. Therefore the synchronization problem can be formulated as a H∞ robust stability problem of the synchronization error system given in Eq. (5.56). To solve this synchronization problem, namely to determine the synchronization control gain matrices Li , it is sufficient to find a feasible solution to the following LMI optimization problem:
October 7, 2008
144
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
minδ,Si ,Sj ,Fi ,V δ S j − V T − V V A i − F i C i V E i − F i Di ˜ ≺ 0, ATi V T − CiT FiT −Si + C˜ T C˜ C˜ T D s.t. T T T T T ˜ T ˜ ˜ ˜ E i V − D i Fi D C −δI + D D (5.58) ∀(i, j) ∈ {1 · · · N }; where the unknowns are the symmetric definite positive matrices Si , Sj ∈ Rn×n , i = 1, . . . , N , j = 1, . . . , N , any matrices V ∈ Rn×n √ and Fi ∈ Rn×r , and the scalar γ , δ > 0. The synchronization gain is given by Li = V −1 Fi . Moreover, the minimum H∞ disturbance attenuation level from the input wk to the weighting synchronization error output ‘zk ’ corresponds to the optimal solution of the minimization problem, i.e., γ = √ δ. Proof details for this result can be found in [24].
5.4.3.2. Experimental Results with Information Transmission A practical implementation of the Chua oscillator to achieve the robust H∞ synchronization is considered. Further, the information transmission probPSfrag replacements lem is investigated in the light of the information transmission via control principle – ITVC discussed in Sec. 5.4.1.1. The information transmission test is used as a performance index to validate the proposed approach for robust synchronization. All the experiments were performed in the laboratory setup PCCHUA described in Sec. 5.2.3. Consider the information transmission scheme exhibited in Fig. 5.27. This scheme shows the Chua’s oscillator circuits on a unidirectional coupling. η(t)
i(t)
y m (t)
+
R id Transmitter + C1
y s (t) -
R
Controller +
C2 L
u1 (t) u2 (t) u3 (t)
id
Receiver + C1
+
L
C2 +
-
Fig. 5.27. Unidirectional communication system. It is interesting to note that the use of the voltage across C1 , which is the x-variable, has been found to be the best scalar variable through which the dynamics of Chua’s circuit can be observed, see Sec. 5.3.1.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
145
Chaos Control for Chua’s Circuits
The information to be transmitted i(t) is injected on the master Chua’s chaotic oscillator (transmitter ) as a perturbation only in the direction of vC1 (t). Mathematically this perturbation corresponds to a modification on the differential equations that govern the motion of the oscillator in the direction of vC1 (t). Then a scalar signal y m (t) is used to carry the information and to serve as a reference signal to allow the synchronization of the slave oscillator (receiver ). This signal can be corrupted by noise interferences η(t) on the transmission channel.
1
y (V)
0.5
0
-0.5
-1
-6
-4
-2
0 x (V)
2
4
6
Fig. 5.28. Typical information perturbed double scroll attractor exhibited by the master oscillator – real data. This figure should be compared to Fig. 5.9, page 108. The information signal is injected using a voltage controlled current source in parallel with capacitor C1 , shown in Fig. 5.6, page 105.
Despite the fact that the information signal must be of very low amplitude in order to keep the master oscillator in the chaotic regime, the oscillator unperturbed dynamical behavior is considerably altered, as can be seen by comparing Fig. 5.9(a), page 108, with Fig. 5.28. The transmitter system can be represented by the set of differential equations
October 7, 2008
146
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
dvC (t) 1 C1 dt C2 dvC2 (t) dt diL (t) L dt id (vC1 ) m y (t)
= =
vC2 (t)−vC1 (t) R vC1 (t)−vC2 (t) R
− id (vC1 (t)) + i(t) + iL (t)
= −vC2 (t) − rL iL (t) = m 0 vC1 +
(5.59)
(m1 −m0 )(|vC1 +Bp |−|vC1 −Bp |) 2
= vC1 (t) + η(t)
where C1 , C2 , L, R, rL, m0 , m1 e Bp are the circuit’s parameters; i(t) is the scalar information signal; y m (t) is the transmitted signal and η(t) is the noise signal on transmission channel. By construction, the receiver oscillator is fully controllable by the controller signals u(t). Analogous to the transmitter the receiver system has its motion governed by the set of differential equations dvC ∗ (t) C1∗ dt1 dvC ∗ (t) C2∗ dt2 ∗ di∗ (t) L Ldt id (vC1∗ ) s y (t)
= =
vC ∗ (t)−vC ∗ (t) 2
1
R∗ vC ∗ (t)−vC ∗ (t) 1
2
R∗
− id (vC1∗ (t)) + u1 (t) + i∗L (t) + u2 (t)
∗ ∗ = −vC2∗ (t) − rL iL (t) + u3 (t)
= m∗0 vC1∗ +
(5.60)
∗ ∗ ∗ (m∗ 1 −m0 )(|vC ∗ +Bp |−|vC ∗ −Bp |) 1
1
2
= vC1∗ (t)
where C1∗ , C2∗ , L∗ , R∗ , rL∗ , m∗0 , m∗1 e Bp∗ are the receiver circuit’s parameters; u1 (t), u2 (t) and u3 (t) are scalar control signals and y s (t) is the receiver output measured signal. The ITVC principle (Sec. 5.4.1.1) states that any controller that guarantees an identical, or quasi-identical, master-slave synchronization can actually perform as a demodulator and thus recovering the transmitted information. In this way, the control signal u1 (t) corresponds to the demodulated information signal i(t). To carry out the proposed approach the parameters of the Chua oscillator (or the communication system) shown in Fig. 5.27 have to be estimated. This was done by using real data acquired from PCChua, together with the Kalman filter like algorithm known as UKF [40], to estimate the parameters shown in Table 5.7.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
147
Chaos Control for Chua’s Circuits Table 5.7. Estimated parameters of the cillator circuit. Parameter Value Parameter R 1673Ω rL C1 30.14µF m0 m1 C2 185.66µF L 52.28H Bp
chap5
PCChua osValue 0Ω −0.365mS −0.801mS 1.74V
Further, as the PCChua is implemented in a digital platform and the approach has been proposed in discrete-time setting, the transmitter Eq. (5.59) and receiver Eq. (5.60) were discretized and the resulting communication scheme similar to the one shown in (Fig. 5.27) (now in the discrete domain) is represented as a robust master-slave synchronization scheme. By applying the discretization approach presented in [59], a piecewise linear representation for the transmitter and receiver systems was obtained that describes the motion of Chua oscillator circuits by means of three linear local models:
Transmitter
m m i wk xk+1 = Ai xk + Bi + |E{z } y m = C xm + D w i k i k k
Receiver
ι(k)
xsk+1 = Ai xsk + Bi + uk y s = C xs i k k
with the following matrices describing the local motion 1 − T /(RC1 ) − T m0 /C1 T /(RC1 ) 0 A1 = T /(RC2 ) 1 − T /(RC2 ) T /C2 0 −T /L 1 − T rL /L
1 − T /(RC1 ) − T m1 /C1 T /(RC1 ) 0 A2 = T /(RC2 ) 1 − T /(RC2 ) T /C2 0 −T /L 1 − T rL /L
1 − T /(RC1 ) − T m0 /C1 T /(RC1 ) 0 A3 = T /(RC2 ) 1 − T /(RC2 ) T /C2 0 −T /L 1 − T rL /L
(5.61)
October 7, 2008
148
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
B1 =
+T Bp (m1 −m0 ) C1
0 0
−T Bp (m1 −m0 ) 0 C1 , B2 = 0 , B3 = 0 0 0
T 0.001 1 Ei = 0 , Ci = 0 , Di = 0.0001 0 0
100 0 ˜i = 0 C˜i = 0 1 0 , D 001 0 By solving the optimization problem in Eq. (5.58) and considering the discretized master-slave synchronization scheme with the parameters shown in Table 5.7, sampling time T = 10 ms, and a unique synchronization gain (Li = G, ∀i ∈ {1, 2, 3}), the following result can be obtained: G = 99.45 × 10−3
2.93 × 10−3
51.50 × 10−9
T
(5.62)
that guarantee a level γ = 9.70 × 10−4 of disturbance attenuation H∞ . By doing Li = G for i ∈ {1, 2, 3} in Eq. (5.58), the implementation of the proposed approach for robust H∞ synchronization of the chaotic systems, represented by piecewise linear models, was simplified and the transmission of index model ‘i’ to the receiver system is no longer necessary. The weighting matrices Ei and Di were defined based on the amplitudes of the information signal i(t) and the noise signal η(t), respectively. The entry term ι(k) = Ei wk assumes an equivalence of the information signal at discrete time instants k, i.e., ι(k) ≡ CT1 i(kT ). In the context of communication, the performance index H∞ -norm appears as a criterium that ensures a better information reconstruction. This argument is based on that the robust H∞ stabilization of the error system Eq. (5.56) tends to minimize the effect of the exogenous entry ‘wk ’ over the weighting synchronization error vector ‘εk ’ and, consequently, the information signal i(t) has its influence minimized over the transmitted signal y m (t). ˜ were conveniently defined in order The weighting matrices C˜ and D to establish the directions and intensities of the weighting synchronization error vector.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
149
Chaos Control for Chua’s Circuits
Notice that synchronization gain G in Eq. (5.62) must be transformed in order to include the effect of the zero-order hold actuators and to generate compensation signals compatible to the receiver system Eq. (5.60). The synchronization gain considered in the PCChua implementation becomes
GP Cchua
2.99 × 10−3 = 0.54 × 10−3 2.69 × 10−3
≡
C1 G1 /T C2 G2 /T LG3 /T
(5.63)
In order to emphasize the effectiveness of the proposed methodology and to validate its implementation two information transmission experiments were performed in the PCChua platform. In the first experiment, a sinusoidal signal i(t) of amplitude of 0.12 mA and frequency of 0.3 Hz is used as the information signal to be transmitted, as shown in Fig. 5.29. The information signal starts to perturb the natural behavior of the transmitter oscillator at 30 sec and stop at 90 sec. 0.2
i(t) (mA)
0.1
0
−0.1
PSfrag replacements −0.2
0
30
Fig. 5.29.
Time (sec)
90
120
[E1] Information signal: i(t) = 0.12sin(2π0.3t) mA.
Figure 5.30 depicts the signal y m (t) transmitted by the master oscillator (transmitter) that serves as a reference signal to the synchronization of the slave oscillator (receiver). The transmission occurs in the time interval from 15 sec to 105 sec. The synchronization error signal ε(t) corresponding to the difference between the transmitted signal y m (t) and the measured slave signal y s (t) is shown in Fig. 5.31. Outside the time span from 15 sec to 105 sec the controller is inactive and the synchronization error signal corresponds to the natural behavior of the free running receiver oscillator.
October 7, 2008
15:34
150
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
8 6
y m (t) (V)
4 2 0 −2 −4
PSfrag replacements
−6 −8
0
15
Time (sec) Fig. 5.30.
105
120
105
120
[E1] Transmitted signal.
6
e(t) (V)
4
PSfrag replacements
2 0 −2 −4 −6
0
15
30
Fig. 5.31.
Time (sec)
90
[E1] Error signal: e(t) = y m (t) − y s (t) V.
Figure 5.32 shows the control action u1 (t) and Fig. 5.33 shows a close comparison between the transmitted information i(t) and the recovered information represented by the control action u1 (t). In the second experiment an information signal i(t) with a more complex behavior is considered. This information signal is the summation of a sinusoidal signal (0.4 Hz), a square signal (1.0 Hz) and a sawtooth signal (1.5 Hz). All signals have the same amplitude, that is, 0.12 mA. Figure 5.34 shows the transmitted signal y m (t). The synchronization error ε(t) signal and the control signal u1 (t) are depicted in Fig. 5.35 and Fig. 5.36, respectively. In Fig. 5.37, it can be readily noticed that the information from 75 sec to 95 sec is recovered via ITVC principle.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
151
Chaos Control for Chua’s Circuits
1.5
u1 (t) (mV)
1
PSfrag replacements
0.5 0 −0.5 −1 −1.5
0
15
30
Fig. 5.32.
90
Time (sec)
105
120
[E1] Control signal: component u1 (t).
5.4.4. Nonlinear State Feedback In this section, two different control structures will be used to control and synchronize connected Chua circuits, in order to send information from the master to the slave oscillator, similarly to what was presented in Sec. 5.4.3.2. However, in this section the control action will be restricted to the first equation in Eq. (5.60); i.e. u2 (t) = u3 (t) = 0; such that dv ∗ (t) ∗ C1 C1 dt C ∗ dvC2∗ (t) 2 dt ∗ di∗ (t) L L dt id (vC1∗ ) s y (t)
= =
vC ∗ (t)−vC ∗ (t) 2
1
R∗ vC ∗ (t)−vC ∗ (t) 1
2
R∗
− id (vC1∗ (t)) + u(t) + i∗L (t)
∗ ∗ = −vC2∗ (t) − rL iL (t)
= m∗0 vC1∗ +
(5.64)
∗ ∗ ∗ ∗ +Bp |−|vC ∗ −Bp |) (m∗ 1 −m0 )(|vC1 1
2
= vC1∗ (t)
where C1∗ , C2∗ , L∗ , R∗ , rL∗ , m∗0 , m∗1 e Bp∗ are the slave system parameters; u(t) is a scalar control signal, and y s (t) is the receiver output measured signal. The objective is to synchronize both oscillators; the master Eq. (5.59), page 146, and the slave Eq. (5.64), by means of the scalar control signal u(t) at the receiver, and to recover the original information signal i(t) as a byproduct of the synchronization process. The design of the following control structures will highlight the tradeoff between control energy, control objective, and controller complexity, as briefly presented in Sec. 5.4.1, and detailed in [12].
October 7, 2008
15:34
152
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
i(t),u1 (t) (mV)
0.2
0.1
0
−0.1
PSfrag replacements −0.2 75
Time (sec)
90
95
(a) Information signal i(t) - black; Control signal u1 (t) - blue
i(t),u1 (t) (mV)
0.2
0.1
0
−0.1
PSfrag replacements −0.2 75
Time (sec)
90
95
(b) Information signal i(t) - black; Control signal filtered u1 (t) - red Fig. 5.33. [E1] Recovered information without filtering (a) and with filtering (b) (Butterworth filter: 2nd order; fc = 2 Hz).
5.4.4.1. Low Energy Adaptive Proportional Controller In this section a nonlinear controller designed with no prior knowledge of the system dynamics is presented. The only consideration is that, from the results obtained in the previous Sec. 5.4.2 and Sec. 5.4.3, it is known that a low energy controller should be implemented (see also Sec. 5.4.1). The proposed controller structure is quite general, as can be verified by the following set of equations: eT (t) = y s (t) − vC1∗ (t), 1 1 2 , kp (t) = kp + kp 1 − 1 + δp e2T (t) u(t) = kp (eT ) eT (t),
(5.65)
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
153
Chaos Control for Chua’s Circuits
8 6
y m (t) (V)
4 2 0 −2 −4
PSfrag replacements
−6 −8
0
15
105
120
105
120
Time (sec) Fig. 5.34.
[E2] Transmitted signal.
6
e(t) (V)
4 2 0 −2
PSfrag replacements
−4 −6
0
15
30
90
Time (sec) Fig. 5.35.
[E2] Error signal: e(t) = y m (t) − y s (t) V.
where kp1 , kp2 and δp are parameters of this adaptive proportional controller, for which the static characteristic curve is shown in Fig. 5.38. The rationale for the control law Eq. (5.65) is that, when the circuits are almost synchronized, the slave oscillator is following a trajectory that is very close to its natural behavior, and therefore a minimum control effort is necessary to keep the synchronization condition. At the same time, as discussed in Sec. 5.4.1.1, a low energy controller should be used as demodulator for master-slave connected nonlinear oscillators, and therefore the proportional gain is reduced when |eT | ≈ 0 (Fig. 5.38), in order to fulfill this requirement. It is important to note that in Eq. (5.65) there are no assumptions about the master oscillator structure and, at a first sight, it is far from obvious that the control action can actually synchronize both Chua circuits, and at
October 7, 2008
15:34
154
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
1.5
u1 (t) (mV)
1
PSfrag replacements
0.5 0 −0.5 −1 −1.5
0
15
30
90
105
120
Time (sec) Fig. 5.36.
[E2] Control signal: component u1 (t).
the same time it will be able to approximate the original information signal after the synchronization takes place. This shows again that the knowledge of the dynamical system model is useful in designing the controllers, but its importance is weighted by the nature of the problem under consideration. Measured results using control Eq. (5.65) are shown in Fig. 5.39 and Fig. 5.40. In Fig. 5.40 one can see that the filtered recovered information signal is worse than those obtained with the H∞ state feedback controller (Sec. 5.4.3.2), as it can be seen by comparison with the result shown in Fig. 5.33, page 152. The slight attenuation of the recovered signal, depicted in Fig. 5.40(b), is mainly related to the strong nonlinear character of Eq. (5.65). Such attenuation becomes more evident during the peaks of the recovered signal. The same phenomenon does not occur with information signals that have smaller amplitudes. In addition, it is worth noting that the delay time between the original and the recovered information signal observed in Fig. 5.40(b), is due to the phase lag introduced by the low-pass Butterworth filter. 5.4.4.2. High Energy Sliding Mode Controller Following the approach of minimal controller complexity of the previous section; i.e. designing a nonlinear controller for Chua’s circuit without relying on accurate knowledge of system model; a simple sliding mode controller will be proposed that is capable of synchronizing two Chua circuits to an arbitrary precision.
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
155
Chaos Control for Chua’s Circuits
i(t),u1 (t) (mV)
0.5
0
PSfrag replacements −0.5 75
90
95
Time (sec) (a) Information signal i(t) - black; Control signal u1 (t) - blue
i(t),u1 (t) (mV)
0.5
0
PSfrag replacements −0.5 75
90
95
Time (sec) (b) Information signal i(t) - black; Control signal filtered u1 (t) - red Fig. 5.37. [E2] Recovered information without filtering (a) and with filtering (b) (Butterworth filter: 2nd order; fc = 10 Hz).
Consider the following set of equations representing a sliding mode controller applied to the slave oscillator Eq. (5.64): eT (t) = y s (t) − vC1∗ (t), umax , eT (t) > δs ; u(t) = umin , eT (t) < −δs ;
(5.66)
where umax = −umin = 0.5mA and δs > 0. Moreover, the control action is switched according to the hysteresis behavior depicted in Fig. 5.41. In this case, by choosing appropriately the value of δs in Eq. (5.66), it is possible synchronize both Chua circuits with arbitrarily small synchronization error. The case δs = 0.4, simulated using normalized equations (Sec. 5.2.1, Eq. (5.4), page 100, is shown in Fig. 5.42. What is relevant in this example is the fact that it is possible to have both oscillators in a quasi-identical synchronization condition and
October 7, 2008
156
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
0.012
0.01
kp
0.008
0.006
0.004
0.002
0 −5
0
5
e [V] T
Fig. 5.38. Adaptive proportional controller characteristic curve for k p1 = 0.0007, kp2 = 0.01, δp = 1.0.
the control action will still be composed only by positive and negative saturation levels umax and umin , no matter the waveform of the information/perturbation signal injected at the transmitter. And this is true even in the case when the strong assumptions of identical vector fields and absence of noise in the transmission channel are considered to be valid. Therefore, despite the fact that the circuit can be made to follow arbitrary reference signals; which includes the synchronization of Chua circuits in master-slave configuration as a special case; the ITVC principle cannot be realized due to the high energy nature of the sliding mode controller. 5.5. Conclusions This chapter has focused on chaos control for Chua circuits. In order to properly address this involved theme, an account of various published works on Chua’s circuit implementation, data analysis, mathematical modeling, dynamical characterization, control and synchronization was presented. Chua’s circuit is undoubtedly a paradigmatic nonlinear oscillator that is capable of exhibiting many interesting dynamical behaviors. In addition, it remains as one of the most robust and easily built chaotic circuits, specially when realized by using simulated inductors (Sec. 5.2.2). With this peculiar construction, one is able to obtain Chua circuits that exhibit very slow oscillations appropriate to digital analysis and control [71]. This path has been followed in the implementation of the so-called PCChua experimental setup (Sec. 5.2.3), which comprises a complete com-
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
157
Chaos Control for Chua’s Circuits
8 6 4
s (V)
2 0 -2 -4 -6 -8 0
10
20
30
40
50
60
70
40
50
60
70
40
50
60
70
t (s)
5 4 3
u (mA)
2 1 0 -1 -2 -3 -4 0
10
20
30 t (s)
6
e = [s - x] (V)
4 2 0 -2 -4 -6 0
10
20
30 t (s)
Fig. 5.39. Synchronization between the perturbed master oscillator and the slave oscillator. Control law as in Eq. (5.65) with kpmin = 0.0007, kpmax = 0.01, δp = 1.0. (a) Transmitted signal s(t). Information signal perturbation starts at t = 30s. (b) Control action at the receiver. Controller activated at t = 15s. (c) Synchronization error [s(t) − y1 (t)].
puter based system flexible enough to investigate chaos control through the realization of any control method that can be represented mathematically by Eq. (5.6), page 104. As a consequence of this flexibility, the PCChua can also be used to simulate parametric variations in Chua’s circuit, so that the system bifurcation diagram can be easily obtained from real data, as shown in [73]. One of the most interesting Chua’s circuit dynamical behaviors – the chaotic regime – which is usually represented by the so-called double scroll attractor; was thoroughly investigated in Sec. 5.3 through different attempts of data analysis and mathematical modeling. As a byproduct of such effort, many dynamical properties associated to the circuit of Chua, and also topological characteristics of the double scroll attractor, were re-
October 7, 2008
158
15:34
World Scientific Review Volume - 9in x 6in
chap5
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
0.4 0.3
u (mA)
0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 25
30
35
40
45 t (s)
50
55
60
65
55
60
65
0.4
i(mA) u(mA) (filtered)
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 25
30
35
40
45
50
t (s)
Fig. 5.40. Recovered information signal. Control law as in Eq. (5.65) with k pmin = 0.0007, kpmax = 0.01, δp = 1.0. (a) Control signal at the receiver. (b) (−) Control signal after applying an order 4 Butterworth low-pass filter (fc = 0.5Hz); (· · · ) Original information signal: i(t) = [0.23 sin(2π(0.1)t)]mA.
vealed such as observability indices, symmetry, higher-order spectral coherence, etc. Concerning the problem of controlling Chua’s circuit, in Sec. 5.4 a broad discussion took place in order to show how intertwined are the control of nonlinear dynamical systems with specific properties of control methods such as control energy, control objective and control law complexity (Sec. 5.4.1). One of the key results of such endeavor is the principle called Information Transmission Via Control – ITVC (Sec. 5.4.1.1), that was extensively used on various control and synchronization strategies discussed in this chapter. Linear and nonlinear state feedback based control techniques, together with robust control and synchronization design methods, were applied to the master-slave configuration of two unidirectionally connected Chua circuits (Sec. 5.4.2, Sec. 5.4.3 and Sec. 5.4.4). These examples were aimed to highlight that in the synchronization of chaotic oscillators; usually viewed as a trajectory tracking control problem; one should consider the interesting fact that to follow a chaotic reference, that is similar to the signal produced
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
chap5
159
Chaos Control for Chua’s Circuits
0.6 0.4
u(t) [mA]
0.2
2δ
0
s
−0.2 −0.4
−3
−2
−1
e [V]
0
1
2
3
T
Fig. 5.41.
Switching law Eq. (5.66).
25
||e||
20
||e||
15
10
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 50
55
60 t (s)
5
65
70
0 0
10
20
30
40
50 t (s)
60
70
80
90
100
Fig. 5.42. Synchronization error norm for the oscillators coupled via switching control law Eq. (5.66). Control activated at t = 35s.
by the uncontrolled nonlinear system, is easier to accomplish than following a simple, but unnatural, sinusoidal signal. It is hoped that this chapter can serve as a starting point for investigating Chua’s circuit dynamical characterization, data analysis, mathematical modeling, control and synchronization. The long list of cited references reflects the authors desire to provide an adequate introduction to these subjects.
October 7, 2008
160
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Acknowledgments The authors are grateful to financial support from Brazilian governmental research-funding agencies Conselho Nacional de Desenvolvimento Cientfico e Tecnol´ ogico – CNPq and Funda¸ca ˜o de Amparo a ` Pesquisa do Estado de Minas Gerais – FAPEMIG.
References [1] L.A. Aguirre, J. Maquet, and C. Letellier, Induced one-parameter bifurcations in identified nonlinear dynamical models, Int. J. Bifurcation and Chaos 12, 1, pp. 135-145, (2002). [2] L.A. Aguirre, A nonlinear correlation function for selecting the delay time in dynamical reconstructions, Phys. Lett. 203A, 2,3, pp. 88-94, (1995). [3] L.A. Aguirre, Recovering map static nonlinearities from chaotic data using dynamical models, Physical D 100, 1,2, pp. 41-57, (1997). [4] L.A. Aguirre, S.B. Bastos, M.A. Alves, and Letellier C, Observability of nonlinear dynamics: normalized indices and a time series approach, Submitted, (2008). [5] L.A. Aguirre and S.A. Billings, Retrieving dynamical invariants from chaotic data using NARMAX models, Int. J. Bifurcation and Chaos 5, 2, pp. 449474, (1995). [6] L.A. Aguirre, U.S. Freitas, C. Letellier, and J. Maquet, Structure-selection techniques applied to continuous-time nonlinear models, Physical D 158, 1-4, pp. 1-18, (2001). [7] L.A. Aguirre, E.C. Furtado, and L.A.B. Tˆ orres, Evaluation of dynamical models: Dissipative synchronization and other techniques, Phys. Rev. E 74, 6, (2006). [8] L.A. Aguirre and C.R.F. J´ acome, Cluster analysis of NARMAX models for signal-dependent systems, IEE Proc. Control Theory Appl. 4, 145, pp. 409414, (1998). [9] L.A. Aguirre, C.G. Rodrigues, and E.M.A.M. Mendes, Nonlinear identification and cluster analysis of chaotic attractors from a real implementation of Chua’s circuit, Int. J. Bifurcation and Chaos 6, 7, pp. 1411-1423, (1997). [10] L.A. Aguirre, B.O.S. Teixeira, and L.A.B. Tˆ orres, Using data-driven discrete-time models and the unscented Kalman filter to estimate unobserved variables of nonlinear systems, Phys. Rev. E 72, 026226, pp. 1-12, (2005). [11] L.A. Aguirre and L.A.B. Tˆ orres, Fixed point stability analysis of Chua’s circuit: A case study with a real circuit, International Journal of Circuits, Systems and Computers 7, 2, pp. 111-115, (1997). [12] L.A. Aguirre and L.A.B. Tˆ orres, Control of nonlinear dynamics: Where do models fit in?, Int. J. Bifurcation and Chaos 10, 3, pp. 667-681, (2000). [13] B.R. Andrievskii and A.L. Fradkov, Control of chaos: Methods and appli-
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
[14] [15]
[16]
[17]
[18]
[19] [20] [21] [22]
[23] [24]
[25] [26]
[27] [28] [29]
[30] [31]
chap5
161
cations. I. Methods, Automation and Remote Control 64, 5, pp. 673-713, (2003). A. Antoniou, Realisation of gyrators using OpAmps and their use in RCactive network synthesis, Proceedings IEE 116, 11, pp. 1838-1850, (1969). M.S. Baptista and I.L. Caldas, Phase-locking and bifurcations of the sinusoidally-driven double scroll circuit, Nonlinear Dynamics 17, 2, pp. 119139, (1998). S.A. Billings and S. Chen, Extended model set, global data and threshold model identification of severaly nonlinear systems, International Journal of Control 50, 5, pp. 1897-1923, (1989). S.A. Billings, S. Chen, and M.J. Korenberg, Identification of mimo nonlinear systems using a forward-regression orthogonal estimator, International Journal of Control 49, 6, pp. 2157-2189, (1989). S.A. Billings and Q.M. Zhu, Rational model identification using an extended least-squares algorithm, International Journal of Control 54, 3, pp. 529-546, (1991). E. Bilotta, P. Pantano, and F. Stranges, A gallery of Chua attractors. Part I, Int. J. Bifurcation and Chaos 17, 1, pp. 1-60, (2007). S. Bocaletti, C. Grebogi, Y.C. Lai, H. Mancini, and D. Maza, The control of chaos: theory and applications, Physics Reports 329, 3, pp. 103-197, (2000). S. Bocaletti, J. Kurths, G. Osipov, D.L. Valadares, and C.S. Zhou, The synchronization of chaotic systems, Physics Reports 366, pp. 1-101, (2002). S. Bowong, F.M.M. Kakmeni, and R. Koina, A new synchronization principle for a class of Lur’e systems with applications in secure communication, Int. J. Bifurcation and Chaos 14, 7, pp. 2477-2491, (2004). O. Calvo and J.H.E. Cartwright, Fuzzy Control of Chaos, Int. J. Bifurcation and Chaos 8, 8, pp. 1743-1747, (1998). C.D. Campos, R.M. Palhares, E.M.A.M. Mendes, L.A.B. Tˆ orres, and L.A. Mozelli, Experimental Results on Chua’s Circuit Robust Synchronization via LMIs, Int. J. Bifurcation and Chaos 17, 9, pp. 3199-3209, (2007). M. Casdagli, S. Eubank, J.D. Farmer, and J. Gibson, State space reconstruction in the presence of noise , Physical D 51, pp. 52-98, (1991). U. C ¸ am and R. Kili¸c, Inductorless realization of nonautonomous MLC chaotic circuit using current-feedback operational amplifiers, Journal of Circuits, Systems and Computers 14, 1, pp. 99-107, (2005). C.-T. Chen, Linear System Theory and Design, 3rd edn. (Oxford Univ. Press, 1998). L.O. Chua, Chua Circuit: An Overview 10 years later, Journal of Circuits, Systems and Computers 4, 2, pp. 117-159, (1994). M.V. Corrˆea, E.M.A.M. Mendes, and L.A. Aguirre, Modeling chaotic dynamics with discrete nonlinear rational models, Int. J. Bifurcation and Chaos 10, 5, pp. 1019-1032, (2000). W.L. Ditto and K. Showalter, Focus Issue: Control and synchronization of chaos, Chaos 7, 4, Guest Editors, (1997). S.T. Dziuban, J.G. Ecker, and M. Kupferschmid, Using deep cuts in an ellipsoid algorithm for nonlinear programming, Mathematical Programming
October 7, 2008
162
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
Study 25, pp. 93-107, (1985) [32] J.P. Eckmann and D. Ruelle, Ergodic-Theory of Chaos and Strange Attractors, Reviews of Modern Physics 57, 3, pp. 61-656, (1985). [33] S. Elgar, B. Vanhoff, L.A. Aguirre, U.S. Freitas, and V. Chandran, HigherOrder Spectra of Nonlinear Polynomial Models for Chua’s Circuit, Int. J. Bifurcation and Chaos 12, 8, pp. 2425-2431, (1998). [34] Z. Galias, New Method for Stabilization of Unstable Periodic Orbits in Chaotic Systems, Int. J. Bifurcation and Chaos 5, 1, pp. 281-295, (1995). [35] G. Ghandi, An improved Chua’s circuit and its use in hyperchaotic circuit, Analog Integrated Circ. and Signal Processing 46, 2, pp. 173-178, (2006). [36] G. Gouesbet and C. Letellier, Global vector field reconstruction by using a multivariate polynomial L2 approximation on nets, Phys. Rev. E 49, 6, pp. 4955-4972, (1994). [37] P. Grassberger and I. Procaccia, Measuring the Strangeness of Strange Attractors, Physica D 9, 1-2, pp. 189-208, (1983). [38] H. Huang, H.X. Li, and J. Zhong, Master-slave synchronization of general Lur’e systems with time-varying delay and parameter uncertainty, Int. J. Bifurcation and Chaos 16, 2, pp. 281-294, (2006). [39] M.-J. Jang, C.-L. Chen, and C.-K. Chen, Sliding mode control of chaos in the cubic Chua’s circuit system, Int. J. Bifurcation and Chaos 12, 6, pp. 1437-1449, (2002). [40] S.J. Julier and J.K. Uhlmann, Unscented filtering and nonlinear estimation, Proceedings of the IEEE 92, 3, pp. 401-422, (2004). [41] M.P. Kennedy, Robust OP Amp realization of Chua’s circuit, Frequenz 46, 3-4, pp. 66-80, (1992). [42] M.P. Kennedy and M.J. Ogorzalek, Special issue on chaos synchronization and control: Theory and applications, IEEE Transactions on Circuits and Systems Part I 44, 10, Guest Editors, (1997). [43] H.K. Khalil, Nonlinear Systems. (Prentice Hall, 2002). [44] A.I. Khibnik, D. Roose, and L.O. Chua, On periodic orbits and homoclinic bifurcations in Chua’s circuit with a smooth nonlinearity, Int. J. Bifurcation and Chaos 3, 2, pp. 363-384, (1993). [45] R. Kili¸c, A comparative study on realization of Chua’s circuit: hybrid realizations of Chua’s circuit combining the circuit topologies proposed for Chua’s diode and inductor elements, Int. J. Bifurcation and Chaos 13, 6, pp. 1475-1493, (2003). [46] R. Kili¸c, On current feedback operational amplifier-based realizations of Chua’s circuit, Circuits Systems and Signal Processing 22, 5, pp. 475-491, (2003). [47] R. Kili¸c, O.G. Saracoglu, and F. Yildirim, A new nonautonomous version of Chua’s circuit: Experimental observations, J. of the Franklin Institute – Engineering and Applied Mathematics 343, 2, pp. 191-203, (2006). [48] M.J. Korenberg, S.A. Billings, Y. Liu, and P.J. McIlroy, Orthogonal parameter estimation algorithm for nonlinear stochastic systems, International Journal of Control 48, 1, pp. 193-210, (1989). [49] I.J. Leontaritis and S.A. Billings, Input-output parametric models for non-
chap5
October 7, 2008
15:34
World Scientific Review Volume - 9in x 6in
Chaos Control for Chua’s Circuits
[50]
[51]
[52] [53]
[54]
[55]
[56] [57] [58] [59] [60]
[61] [62]
[63] [64] [65] [66] [67]
chap5
163
linear systems part II: stochastic nonlinear systems, International Journal of Control 41, 2, pp. 329-344, (1985). C. Letellier and L.A. Aguirre, Investigating nonlinear dynamics from time series: the influence of symmetries and the choice of observables, Chaos 12, 3, pp. 549-558, (2002). C. Letellier, L.A. Aguirre, and J. Maquet, Relation between observability and differential embeddings for nonlinear dynamics, Phys. Rev. E 71, 066213, pp. 1-8, (2005). C. Letellier and R. Gilmore, Covering dynamical systems: Two-fold covers , Phys. Rev. E (in press), (2000). C. Letellier and G. Gouesbet, Topological characterization of reconstructed attractors modding out symmetries, Journal de Physique II 6, pp. 16151638, (1996). C. Letellier, G. Gouesbet and N. Rulkov, Topological analysis of chaos in equivariant electronic circuits, Int. J. Bifurcation and Chaos 6, 12, pp. 25312555, (1996). T.-L. Liao and F.-W. Chen, Control of Chua’s circuit with a cubic nonlinearity via nonlinear linearization technique, Circuits, Systems and Signal Processing 17, 6, pp. 719-731, (1998). E. Lorenz, Deterministic Nonperiodic Flows, Journal of Atmospheric Sciences 20, pp. 131-140, (1963). R.A. Madan, Special issue on Chua’s circuit: a paradigm for chaos, Journal of Circuits, Systems and Computers 3, 1, (1993) T. Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Transactions on Circuits and Systems Part I 33, 12, pp. 1055-1058, (1984). E.M.A.M. Mendes and S.A. Billings, A note on discretization of nonlinear differential equations, Chaos 12, 1, pp. 66-71, (2002). L.A. Mozelli, C.D. Campos, R.M. Palhares, L.A.B. Tˆ orres, and E.M.A.M. Mendes, Chaotic synchronization and information transmission experiments: A fuzzy relaxed H∞ control approach, Circuits, Systems and Signal Processing 26, 4, pp. 427-449, (2007). E. Ott, C. Grebogi, and J.A. Yorke, Controlling Chaos, Physical Review Letters 64, 11, pp. 1196-1199, (1990). A.G. Radwan, A.L. Soliman, and A.M. El-Sedeek, An inductorless CMOS realization of Chua’s circuit, Chaos, Solitons and Fractals 18, pp. 149-158, (2003). D. Ruelle and F. Takens, Nature of Turbulence, Communications in Mathematical Physics 23, 4, (1971). S. Sastry, Nonlinear Systems: analysis, stability and control. (SpringerVerlag New York, Inc., 1999). A.S. Sedra and K.C. Smith, Microelectronic Circuits. (Oxford University Press, USA, 2003). J.-J. Slotine and W. Li, Applied Nonlinear Control. (Prentice Hall, 1990). F.O. Souza, R.M. Palhares, E.M.A.M. Mendes, and L.A.B. Tˆ orres, Robust H∞ control for master-slave synchronization of Lur’e systems with timedelay feedback control, Int. J. Bifurcation and Chaos To appear, (2008).
October 7, 2008
164
15:34
World Scientific Review Volume - 9in x 6in
L.A.B. Tˆ orres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes
[68] J.A.K. Suykens, P.F. Curran, and L.O. Chua, Robust synthesis for masterslave synchronization of Lur’e systems, IEEE Transactions on Circuits and Systems Part I 46, 7, pp. 841-850, (1999). [69] L.A.B. Tˆ orres, Discrete-time dynamic systems synchronization: Information transmission and model matching, Physica D 228, 1, pp. 31-39, (2007). [70] L.A.B. Tˆ orres and L.A. Aguirre, An extended chaos control method applied to the Chua’s circuit, Electronics Letters 35, 10, pp. 768-770, (1999). [71] L.A.B. Tˆ orres and L.A. Aguirre, Inductorless Chua’s circuit, Electronics Letters 36, 23, pp. 1915-1916, (2000). [72] L.A.B. Tˆ orres and L.A. Aguirre, Transmitting information by controlling nonlinear oscillators, Physica D 196, 3-4, pp. 387-406, (2004). [73] L.A.B. Tˆ orres and L.A. Aguirre, PCCHUA – A laboratory setup for relatime control and synchronization of chaotic oscillations, Int. J. Bifurcation and Chaos 15, 8, pp. 2349-2360, (2005). [74] T. Wu and M.-S. Chen, Chaos control of modified Chua’s System, Physica D 164, pp. 53-58, (2002). [75] Q.M. Zhu and S.A. Billings, Recursive parameter estimation for nonlinear rational models, J. Syst. Eng. 1, pp. 63-76, (1991). [76] Q.M. Zhu and S.A. Billings, Fast ortogonal identification of nonlinear stochastic models and radial basis functions, International Journal of Control 64, 5, pp. 871-886, (1996).
chap5
Chapter 6 CHAOS CONTROL FOR A PWM H-BRIDGE INVERTER
B. Robert* and M. Feki† Department of Electronics, Electrotechnics and Automatic, University of Reims-Champagne-Ardenne, France. * [email protected] † [email protected] H.H.C. Iu School of Electrical, Electronic and Computer Engineering, The University of Western Australia, Perth, Australia. [email protected] Pulse width modulation (PWM) current-mode single phase inverters are known to exhibit bifurcations and chaos when parameters vary or if the gain of the proportional controller is arbitrarily increased. Our aim in this chapter is to show, using control theory and numerical simulations, how to apply a method to stabilize the desired periodic orbit for larger values of the proportional gain. To accomplish this aim, a time-delayed feedback controller (TDFC) is used in conjunction with the proportional controller in its simple form as well as in its extended form (ETDFC). The main advantages of those methods are the robustness and ease of construction because they do not require the knowledge of an accurate model but only the period of the target unstable periodic orbit (UPO). Moreover, to improve the dynamical performances, an optimal criterion and an adaptive law are defined to determine the control parameters.
6.1. Introduction Power electronics is a discipline that has emerged from the need to convert electrical energy. Its field of application is wide and concerns industrial, commercial, residential and also aerospace environments. 165
166
B. Robert, M. Feki and H.H.C. Iu
Power converters are basic switching circuits that are modeled by a number of linear differential equations corresponding to different topologies. The toggling between different topologies can either be done naturally due to switching characteristics or done under the force of a feedback control system. Due to the existence of various operating modes and control saturations, the overall operation is compared to a piecewise smooth nonlinear dynamical system. Early studies by Hamill and coworkers [6, 11] have shown that switching power converters can exhibit several nonlinear phenomena. These include subharmonic oscillations, quasi-periodic operations, bifurcations and chaos. Chaos in power electronics have been observed by many researchers [8, 11, 19]. In fact, engineers have frequently encountered chaos in power electronics systems, but more often than not this phenomenon was considered as strange and undesirable, hence engineers usually attempted to avoid chaos. During the last two decades, tools of analyzing bifurcations and chaos have been well developed. Therefore, the investigation of nonlinear dynamics in power electronics has become popular. Actually, much work has focused on bifurcation and chaos due to parameter variations [1, 4, 7]. Recently it has been shown by [19] that control parameters themselves may lead to bifurcation and chaos if they are not properly selected. Since the seminal paper [15] (OGY), control of chaos has been the focus of a growing literature. Knowing that a chaotic attractor contains infinitely many UPO, the OGY methods take advantage of the great sensitivity of chaotic orbits to stabilize a UPO by appropriately perturbing an accessible parameter [9, 10]. These methods suffer from lack of robustness to imprecise measurements and to uncontrolled parameter variations. To overcome this deficiency, the TDFC has been proposed as an alternative method [16, 17]. TDFC is known for its robustness and simplicity of construction. Besides, the system model need not be known but only the length τ of the UPO to be stabilized is essential. The control signal is proportional to the difference between the current state and the τ -delayed state. Once the desired UPO is stabilized the control signal vanishes. This method has been successfully applied to the control of
Chaos Control for a PWM H-Bridge Inverter
167
duffing equations [2], discrete chaotic systems [14] and to the control of power converters [3, 12, 20]. In this chapter we propose to control a PWM current-mode H-bridge inverter using TDFC [13] and ETDFC methods in conjunction with the proportional controller. A discrete model describing the behavior of the converter is developed and digital controllers are designed. Using the Jury criteria a stability zone in the parameter space is defined. In this study we show the effect of the TDFC and ETDFC on the bifurcation diagrams and we present two-dimensional bifurcation diagrams. Our work will be outlined as follows. In section 6.2 we present the discrete model of an H-bridge converter circuit. Section 6.3 gives an overview on the effect of the variation of the proportional gain and boosting of the chaotic behavior. Sections 6.4 and 6.5 will be devoted to the presentation of the TDFC and ETDFC controllers and their results, respectively. In section 6.4 we show a two-degree of freedom design procedure and the resulting two-dimensional bifurcation diagram. The ETDFC controller presented in section 6.5 leads to a three-degrees of freedom design and improves the results obtained in the preceding sections. Section 6.6 includes results on sinusoidal output tracking. Finally, our conclusions and remarks are stated in section 6.7. 6.2. H-Bridge Model In order to increase dynamical performances, a growing number of applications in the field of electrical engineering require that they be fed by a precise current generator. However, all industrial power sources are voltage generators. It therefore follows that converters are necessary to adapt voltage sources to loads. Indeed, owing to the presence of many windings in electrical machines, most of them are naturally inductive and are deemed to be current sources. A convenient way to adapt sources is to add a current control to a voltage converter. In this section, we describe the converter structure and its running mode before setting up its sampled data model. With the goal to increase the efficiency of the power stages, static converters operate by switching the load between several voltage sources. There is a wide range of conversion structures, more or less
B. Robert, M. Feki and H.H.C. Iu
168
complex, whose choice depends on many parameters. By controlling the switching pattern over the operating period, it is possible to vary the average output voltage. This technique, called Pulse Width Modulation (PWM), is very widespread. A well-known structure, used in most variable speed drives, is the HBridge. Fig. 6.1 illustrates the circuit of this voltage inverter i.e. a DC/AC converter. A switch is realized in this bridge by combining a bipolar transistor and an anti-parallel diode. The inverter is fed by a voltage source E and it supplies a resistive and inductive load (L, R). The output current is controlled by a current loop. The four equivalent switches, named S1, S2, S3 and S4, are shared among two pairs (S1, S2) and (S3, S4). Pairs are controlled by the PWM modulator in a complementary way. States of the switches define two distinct topologies of the inverter, T1 (S1, S2 on and S3, S4 off) and T2 (S1, S2 off and S3, S4 on), and yield to two opposite voltages across the load. For example, topology T1 implies that the voltage across the load v = E.
S1
S3 L
E
R
i v S4
S2
Fig. 6.1. H-bridge. The PWM modulator defines the switching period T. It generates a high state (C = 1), centered in the period, whose duration is defined by the current controller acting on the duty ratio d. dn denotes the duty cycle on the nth functioning period. The low state (C = 0) is split into two parts at the beginning and the end of the period. So the normal running mode
Chaos Control for a PWM H-Bridge Inverter
169
of the converter involves a sequence of three topologies: T2 −T1 −T2. Then the voltage v evolves as depicted in Fig. 6.2 and v = −(−1)CE. This pattern avoids switchings and interferences at the sampling times because the digital controller samples current at the beginning of the switching period. At that point the sampling frequency is equal to the fundamental PWM frequency.
Fig. 6.2. Centered PWM voltage and current waveform. Fig. 6.2 also shows different possible waveforms of the output current under a normal periodic operating mode. As explain before, it is important to reduce the current ripple within a period. For that aim (see the dashed curve), the inverter period is chosen much smaller than the electric time constant L/R. This figure assumes also that the reference current In is constant. In fact, it is possible to track a sinusoidal current reference (see section 6.6). Due to the sampled nature of the current controller, a nonlinear map will be required as a model of the converter instead of a usual linear averaged state space model. For the sake of generality, the current is
170
B. Robert, M. Feki and H.H.C. Iu
scaled with respect to the maximum output current E/R and the voltage is scaled to E. On each interval, the one-dimensional model is described by a single linear differential equation.
−
1 di = i + (−1) C , 2δ dt
(6.1)
where δ = RT/2L << 1 in order to reduce the ripple current. By integrating on the three intervals and stacking up the solutions, the model becomes a nonlinear discrete time map.
in +1 = αin + β (2 sinh(δd n ) − sinh(δ )),
(6.2)
where α = e−2δ and β = 2e−δ. For the sake of clarity, it is important to note that due to boundedness of the duty cycle d, the PWM modulator introduces saturations to limit too much wandering i.e. to keep the duty cycle d inside the interval [0, 1]. Saturated running modes are characterized by a unique topology over T namely T1 or T2. Saturation of the modulator requires in fact a piecewise model and can be studied in an analytical way by the normal form theory. Saturations lead to the emergence of border collision bifurcations when controller parameters vary. These phenomena have been thoroughly investigated in previous works [18, 19]. Because our goal in this chapter is to define a new controller stabilizing the normal running mode over a wide range, we focus on the standard model (6.2). With the objective of the design of an experimental prototype, the constants are chosen in order to make it easier with a power that does not exceed a few kW and to satisfy the frequency condition presented above. To maintain the ripple current at a low level, we set L/R = 0.5ms and T = 0.2ms, hence parameters were chosen as follows: R = 40Ω, L = 20mH, E = 400V, leading to the following constants: δ = 0.2, α = 670.32 × 10−3, β = 1.6375.
Chaos Control for a PWM H-Bridge Inverter
171
6.3. Current-programmed Inverter Fig. 6.3 depicts an H-Bridge current-programmed single phase inverter. The proportional corrector is the main controller used to control the switching process and is given by:
γ n = γ Pn = k ( I n − in ),
(6.3)
where k is a proportional gain and In is the reference current. In this case, the PWM modulator generates the duty cycle:
dn =
1 1 + γ Pn = + k ( I n − in ). 2 2
(6.4)
Clock:T S3
S1
In
Controller
γn
i
in
Digital Current Control
v
Current sensor
PWM Modulator
S4
E L
R
S2
Fig. 6.3. Current-programmed single phase inverter. With the objective of the design of an experimental prototype, the constants are chosen in order to make it easier with a power that does not exceed a few kW and to satisfy the frequency condition presented above. To maintain the ripple current at a low level, we set L/R = 0.5ms and T = 0.2ms, hence parameters were chosen as follows: As a matter of fact, when static converters are driven by a T-periodic clock, it is important to have a T-periodic output current. Using the discrete-time map given by (6.2) and (6.4), the T-periodic output corresponds to a fixed point of order one. The local stability of the T-periodic mode is analyzed by deriving the Jacobian of the discrete-time map in the neighborhood of the
172
B. Robert, M. Feki and H.H.C. Iu
Fig. 6.4. Bifurcation diagram with proportional corrector. fixed points. By solving in+1 = in we obtain the fixed point denoted by i∗ as a function of k, and thereby we can determine the loci of the eigenvalues when k varies. By combining the inverter model (6.2) and the proportional controller (6.4) we obtain a first-order closed loop iteration with a unique eigenvalue λP given by:
λ P = α − 2βδk cosh(δd * ),
(6.5)
where d∗ is the solution of:
⎧(α − 1)i * + β (2 sinh(δd * ) − sinh(δ )) = 0, ⎪ 1 ⎨ d * = k ( I n − i * ). ⎪⎩ 2
(6.6)
Chaos Control for a PWM H-Bridge Inverter
173
When k varies the eigenvalue crosses the unit circle at λ0 = −1 with k = k0. Using the fact that δd∗ <<1 it follows that cosh(δd∗ ) ≈ 1 and
k0 ≈
α +1 = 2.55. 2 βδ
(6.7)
Therefore, for k > k0 the T-periodic mode becomes unstable. The bifurcation diagram depicted in Fig. 6.4 was obtained by iterating (6.2) and (6.4) and the exact value of k0 = 2.52 is obtained. It has been shown in [19] that the fixed point of order one (corresponding to T-periodic orbit) continues to exist but becomes unstable. 6.4. Time-delayed Feedback Controller 6.4.1. Controller Design The TDFC controller is similar to a proportional corrector but referenced to the same state delayed by a time τ equal to the length of the UPO to be stabilized. The aim of this section is to stabilize the UPO of length T. Using the discrete-time notation, this corresponds to stabilizing the fixed point of order one for higher values of k thus the TDFC expression is given by:
γ Dn = η (in − in −1 ),
(6.8)
where η is the TDFC gain to be adjusted. Fig. 6.5 shows the block diagram of the TDFC controller. The whole control signal and the duty cycle become:
γ n = γ Pn + γ Dn , dn =
1 + k ( I n − in ) + η (in − in −1 ). 2
(6.9)
B. Robert, M. Feki and H.H.C. Iu
174
Clock.T Sampled Digital Control εn
I
k
γn
γPn
PWM
in γDn Delay
in-1
η
Fig. 6.5. TDFC controller.
Fig. 6.6. Stability zone in the k-η space. 6.4.2. Stability Analysis When TDFC is applied, the discrete system defined by (6.2) and (6.8) becomes second order since it involves in and in−1. Let us define xn = in−1 − i∗ and yn = in − i∗ , using xn and yn as the new state variables and combining (6.2) and (6.8) yields
Chaos Control for a PWM H-Bridge Inverter
175
x n +1 = y n , y n +1 = αy n + (α − 1)i + β (2 sinh(δd n ) − sinh(δ )). *
(6.10)
We investigate the local stability by deriving the Jacobian evaluated at the origin:
0 1 ⎡ ⎤ . J (d * ) = ⎢ * * ⎥ ⎣ J 1 (η , d ) J 2 (η , k , d )⎦
(6.11)
Since at equilibrium we have in = in−1 then TDFC signal is null. Thus d∗ is again the solution of (6.6) and it follows that
J 1 (η , d * ) = −2βηδ cosh(δd * ), J 2 (η , k , d * ) = α + 2βδ (η − k ) cosh(δd * ). The characteristic equation is given by:
z 2 − J 2 z − J 1 = 0.
(6.12)
The Jury criterion is used to evaluate the stability domain in terms of k and η. According to the Jury criterion, the system is stable if the following conditions are satisfied:
− J 1 < 1,
(6.13)
1 − J 2 − J 1 > 0,
(6.14)
1 + J 2 − J 1 > 0.
(6.15)
(6.13) yields to an upper limit on η
η<
1 , 2βδ cosh(δd * )
(6.16)
176
B. Robert, M. Feki and H.H.C. Iu
(6.14) yields to a lower limit on k
k>
α −1 2βδ cosh(δd * )
.
(6.17)
We should note that α − 1 < 0, then it is sufficient to choose k > 0, in order to satisfy condition (6.17) as well as the positiveness of the proportional corrector gain. The last of Jury’s conditions (6.15) leads to an affine relation between η and k:
η>
k α +1 − . 2 4βδ cosh(δd * )
(6.18)
Using conditions (6.16)–(6.18), a stability zone in the parameters space k − η is defined and depicted in Fig. 6.6. The shaded zone has a special interest if we assume that the reference is constant. Indeed, (6.9) can be written in the following form:
γ n = k ( I n − in ) − η (( I n − in ) − ( I n − in −1 )). Denote εn = In−in the steady state error, then the control signal is similar to a Proportional-Derivative (P.D.) controller:
γ n = K P ε n + K D (ε n − ε n −1 ), where KP = k and KD = −η. Ordinarily we have KP > 0 and KD < 0, then the conjunction of the proportional and the TDFC controllers can be assimilated to a P.D. controller inside the shaded zone of Fig. 6.6. It is known from linear control theory that the larger the proportional gain k the less is the steady state error ε. In view of Fig. 6.6, k is maximized when η is maximized and the limit of stability is the point M which is the intersection of boundary lines
L1 : η =
1 2βδ
and L2 : η =
k 1+α − 2 4βδ
Chaos Control for a PWM H-Bridge Inverter
177
obtained from (6.16) and (6.18) with the approximation cosh(δd∗ )≈ 1. The coordinates of M are:
η max ≈ k max ≈
1 2βδ
= 1.5,
3+α = 5.6. 2βδ
(6.19)
(6.20)
6.4.3. Results The two-dimensional bifurcation diagram depicted in Fig. 6.7 indicates in brown the zone of the period one mode and this matches the stability zone shown in Fig. 6.6. Zones of higher periods as well as chaotic zones are also shown as parameters k and η vary. Particularly, when η = 0 we observe that as k increases the output current is of period one then becomes period two and for a short interval of k becomes period four before it goes into chaotic mode. As k increases further the output becomes period six and finally arrives at period three mode and this matches very well with the one-dimensional bifurcation diagram shown in Fig. 6.4. We also notice that as η increases and a TDFC controller is acting then the period one mode extends for higher values of k as expected (see also Fig. 6.8). Fig. 6.8 shows that the fixed point of order one that was embedded in a zone of period two mode (2.5 ≤ k ≤ 3.3) and in a chaotic attractor (3.3 ≤ k ≤ 5.0) has been stabilized by the application of the TDFC in conjunction with the proportional controller. 6.4.4. Optimality Criterion The main concern of the previous design was to obtain stability of the fixed point of order one for larger values of k which reduces the steady state error εn. In this section, we show that a drawback of TDFC is to degrade the dynamical response by increasing the settling time. Fig. 6.9 shows the sampled and hold output current resulting from a step reference from In = 0 to In = 0.5 with k = 2 and different values of η. We
B. Robert, M. Feki and H.H.C. Iu
178
note that the transient time is increased when the TDFC is applied. When η tends to ηmax, the transient time becomes too large. To overcome this problem we present an adaptive law to calculate a value of η that leads to a fast response for each value of k. The linearization of system (6.10) gives
⎡ x n +1 ⎤ ⎢y ⎥ = ⎣ n +1 ⎦
⎡x ⎤ ⎡x ⎤ J ⎢ n ⎥ = J n +1 ⎢ 0 ⎥. ⎣ yn ⎦ ⎣ y0 ⎦
This implies that [5]
x n +1 y n +1
= J n +1
x0 y0
�